Dirichlet series
| L(s) = 1 | − 6·2-s + 18·4-s − 6·5-s + 6·7-s − 39·8-s − 12·9-s + 36·10-s − 18·11-s + 21·13-s − 36·14-s + 60·16-s − 3·17-s + 72·18-s − 24·19-s − 108·20-s + 108·22-s − 102·23-s − 60·25-s − 126·26-s + 18·27-s + 108·28-s + 147·29-s + 99·31-s + 3·32-s + 18·34-s − 36·35-s − 216·36-s + ⋯ |
| L(s) = 1 | − 3·2-s + 9/2·4-s − 6/5·5-s + 6/7·7-s − 4.87·8-s − 4/3·9-s + 18/5·10-s − 1.63·11-s + 1.61·13-s − 2.57·14-s + 15/4·16-s − 0.176·17-s + 4·18-s − 1.26·19-s − 5.39·20-s + 4.90·22-s − 4.43·23-s − 2.39·25-s − 4.84·26-s + 2/3·27-s + 27/7·28-s + 5.06·29-s + 3.19·31-s + 3/32·32-s + 9/17·34-s − 1.02·35-s − 6·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.000370735\) |
| Root analytic conductor: | \(0.719522\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 19^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.03408815008\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03408815008\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 19 | \( 1 + 24 T - 234 T^{2} - 1182 p T^{3} - 774 p^{2} T^{4} + 726 p^{3} T^{5} + 1583 p^{4} T^{6} + 726 p^{5} T^{7} - 774 p^{6} T^{8} - 1182 p^{7} T^{9} - 234 p^{8} T^{10} + 24 p^{10} T^{11} + p^{12} T^{12} \) |
| good | 2 | \( 1 + 3 p T + 9 p T^{2} + 39 T^{3} + 21 p^{2} T^{4} + 141 T^{5} + 29 p T^{6} - 9 p^{5} T^{7} - 627 T^{8} - 1113 T^{9} - 1041 p T^{10} + 27 T^{11} + 7077 T^{12} + 27 p^{2} T^{13} - 1041 p^{5} T^{14} - 1113 p^{6} T^{15} - 627 p^{8} T^{16} - 9 p^{15} T^{17} + 29 p^{13} T^{18} + 141 p^{14} T^{19} + 21 p^{18} T^{20} + 39 p^{18} T^{21} + 9 p^{21} T^{22} + 3 p^{23} T^{23} + p^{24} T^{24} \) |
| 3 | \( 1 + 4 p T^{2} - 2 p^{2} T^{3} + 8 p T^{4} - 34 p^{2} T^{5} + 134 T^{6} - 373 p^{2} T^{7} + 730 p T^{8} - 2752 p^{2} T^{9} + 12719 p T^{10} + 7339 p^{2} T^{11} + 718057 T^{12} + 7339 p^{4} T^{13} + 12719 p^{5} T^{14} - 2752 p^{8} T^{15} + 730 p^{9} T^{16} - 373 p^{12} T^{17} + 134 p^{12} T^{18} - 34 p^{16} T^{19} + 8 p^{17} T^{20} - 2 p^{20} T^{21} + 4 p^{21} T^{22} + p^{24} T^{24} \) | |
| 5 | \( 1 + 6 T + 96 T^{2} + 467 T^{3} + 1137 p T^{4} + 26151 T^{5} + 10174 p^{2} T^{6} + 1092051 T^{7} + 9452817 T^{8} + 38413514 T^{9} + 297623811 T^{10} + 1122741453 T^{11} + 8011385341 T^{12} + 1122741453 p^{2} T^{13} + 297623811 p^{4} T^{14} + 38413514 p^{6} T^{15} + 9452817 p^{8} T^{16} + 1092051 p^{10} T^{17} + 10174 p^{14} T^{18} + 26151 p^{14} T^{19} + 1137 p^{17} T^{20} + 467 p^{18} T^{21} + 96 p^{20} T^{22} + 6 p^{22} T^{23} + p^{24} T^{24} \) | |
| 7 | \( 1 - 6 T - 117 T^{2} + 608 T^{3} + 8154 T^{4} - 29562 T^{5} - 205826 T^{6} - 896976 T^{7} - 2284875 T^{8} + 114186218 T^{9} + 743881389 T^{10} - 3902118156 T^{11} - 38601269423 T^{12} - 3902118156 p^{2} T^{13} + 743881389 p^{4} T^{14} + 114186218 p^{6} T^{15} - 2284875 p^{8} T^{16} - 896976 p^{10} T^{17} - 205826 p^{12} T^{18} - 29562 p^{14} T^{19} + 8154 p^{16} T^{20} + 608 p^{18} T^{21} - 117 p^{20} T^{22} - 6 p^{22} T^{23} + p^{24} T^{24} \) | |
| 11 | \( 1 + 18 T - 228 T^{2} - 5552 T^{3} + 29166 T^{4} + 992034 T^{5} - 267118 p T^{6} - 164354796 T^{7} - 362171790 T^{8} + 16473984520 T^{9} + 121721077314 T^{10} - 623845748532 T^{11} - 15282031103345 T^{12} - 623845748532 p^{2} T^{13} + 121721077314 p^{4} T^{14} + 16473984520 p^{6} T^{15} - 362171790 p^{8} T^{16} - 164354796 p^{10} T^{17} - 267118 p^{13} T^{18} + 992034 p^{14} T^{19} + 29166 p^{16} T^{20} - 5552 p^{18} T^{21} - 228 p^{20} T^{22} + 18 p^{22} T^{23} + p^{24} T^{24} \) | |
| 13 | \( 1 - 21 T + 114 T^{2} - 3048 T^{3} + 99153 T^{4} - 1219683 T^{5} + 6059104 T^{6} - 140861934 T^{7} + 4074921219 T^{8} - 2620355208 p T^{9} + 57415349265 T^{10} - 4773855887766 T^{11} + 121298781358881 T^{12} - 4773855887766 p^{2} T^{13} + 57415349265 p^{4} T^{14} - 2620355208 p^{7} T^{15} + 4074921219 p^{8} T^{16} - 140861934 p^{10} T^{17} + 6059104 p^{12} T^{18} - 1219683 p^{14} T^{19} + 99153 p^{16} T^{20} - 3048 p^{18} T^{21} + 114 p^{20} T^{22} - 21 p^{22} T^{23} + p^{24} T^{24} \) | |
| 17 | \( 1 + 3 T - 84 T^{2} + 3702 T^{3} - 39603 T^{4} - 812499 T^{5} + 4691260 T^{6} - 149973228 T^{7} - 3075033123 T^{8} + 136201755708 T^{9} - 454077765279 T^{10} - 22107577523112 T^{11} + 875532343794093 T^{12} - 22107577523112 p^{2} T^{13} - 454077765279 p^{4} T^{14} + 136201755708 p^{6} T^{15} - 3075033123 p^{8} T^{16} - 149973228 p^{10} T^{17} + 4691260 p^{12} T^{18} - 812499 p^{14} T^{19} - 39603 p^{16} T^{20} + 3702 p^{18} T^{21} - 84 p^{20} T^{22} + 3 p^{22} T^{23} + p^{24} T^{24} \) | |
| 23 | \( 1 + 102 T + 5880 T^{2} + 242044 T^{3} + 8074740 T^{4} + 232105944 T^{5} + 5933501770 T^{6} + 135098714970 T^{7} + 2764757638548 T^{8} + 51583388614096 T^{9} + 904612835742672 T^{10} + 15789113035371534 T^{11} + 320785786069630231 T^{12} + 15789113035371534 p^{2} T^{13} + 904612835742672 p^{4} T^{14} + 51583388614096 p^{6} T^{15} + 2764757638548 p^{8} T^{16} + 135098714970 p^{10} T^{17} + 5933501770 p^{12} T^{18} + 232105944 p^{14} T^{19} + 8074740 p^{16} T^{20} + 242044 p^{18} T^{21} + 5880 p^{20} T^{22} + 102 p^{22} T^{23} + p^{24} T^{24} \) | |
| 29 | \( 1 - 147 T + 10638 T^{2} - 548808 T^{3} + 23661696 T^{4} - 919992729 T^{5} + 33943669042 T^{6} - 1192831592220 T^{7} + 39785563065882 T^{8} - 1286899558576728 T^{9} + 40759217279931174 T^{10} - 1260467388512258220 T^{11} + 37512869403908106759 T^{12} - 1260467388512258220 p^{2} T^{13} + 40759217279931174 p^{4} T^{14} - 1286899558576728 p^{6} T^{15} + 39785563065882 p^{8} T^{16} - 1192831592220 p^{10} T^{17} + 33943669042 p^{12} T^{18} - 919992729 p^{14} T^{19} + 23661696 p^{16} T^{20} - 548808 p^{18} T^{21} + 10638 p^{20} T^{22} - 147 p^{22} T^{23} + p^{24} T^{24} \) | |
| 31 | \( 1 - 99 T + 8727 T^{2} - 540540 T^{3} + 30402945 T^{4} - 1458608319 T^{5} + 64909332043 T^{6} - 2619622606347 T^{7} + 99818681518848 T^{8} - 3553487633552742 T^{9} + 121725452021726463 T^{10} - 3963361277030280903 T^{11} + \)\(12\!\cdots\!15\)\( T^{12} - 3963361277030280903 p^{2} T^{13} + 121725452021726463 p^{4} T^{14} - 3553487633552742 p^{6} T^{15} + 99818681518848 p^{8} T^{16} - 2619622606347 p^{10} T^{17} + 64909332043 p^{12} T^{18} - 1458608319 p^{14} T^{19} + 30402945 p^{16} T^{20} - 540540 p^{18} T^{21} + 8727 p^{20} T^{22} - 99 p^{22} T^{23} + p^{24} T^{24} \) | |
| 37 | \( 1 - 8874 T^{2} + 39453723 T^{4} - 117970083763 T^{6} + 266195955119445 T^{8} - 481289917598888787 T^{10} + \)\(72\!\cdots\!18\)\( T^{12} - 481289917598888787 p^{4} T^{14} + 266195955119445 p^{8} T^{16} - 117970083763 p^{12} T^{18} + 39453723 p^{16} T^{20} - 8874 p^{20} T^{22} + p^{24} T^{24} \) | |
| 41 | \( 1 + 144 T + 11163 T^{2} + 355752 T^{3} - 8464866 T^{4} - 1542494952 T^{5} - 62169863144 T^{6} + 399495011904 T^{7} + 176879623935177 T^{8} + 8978616925749936 T^{9} + 138472442393345793 T^{10} - 8947832283253483704 T^{11} - \)\(65\!\cdots\!47\)\( T^{12} - 8947832283253483704 p^{2} T^{13} + 138472442393345793 p^{4} T^{14} + 8978616925749936 p^{6} T^{15} + 176879623935177 p^{8} T^{16} + 399495011904 p^{10} T^{17} - 62169863144 p^{12} T^{18} - 1542494952 p^{14} T^{19} - 8464866 p^{16} T^{20} + 355752 p^{18} T^{21} + 11163 p^{20} T^{22} + 144 p^{22} T^{23} + p^{24} T^{24} \) | |
| 43 | \( 1 + 27 T + 1068 T^{2} - 61858 T^{3} + 227013 T^{4} + 26711865 T^{5} + 3468305848 T^{6} + 113462971236 T^{7} + 2290399946613 T^{8} - 31890759747304 T^{9} - 13892532646710483 T^{10} - 130352073354404160 T^{11} + 1800910079507940253 T^{12} - 130352073354404160 p^{2} T^{13} - 13892532646710483 p^{4} T^{14} - 31890759747304 p^{6} T^{15} + 2290399946613 p^{8} T^{16} + 113462971236 p^{10} T^{17} + 3468305848 p^{12} T^{18} + 26711865 p^{14} T^{19} + 227013 p^{16} T^{20} - 61858 p^{18} T^{21} + 1068 p^{20} T^{22} + 27 p^{22} T^{23} + p^{24} T^{24} \) | |
| 47 | \( 1 + 99 T + 6018 T^{2} + 294074 T^{3} + 10863897 T^{4} + 565964145 T^{5} + 25757282596 T^{6} + 583337737242 T^{7} + 10177397826039 T^{8} - 1186851009182872 T^{9} - 69183145856627091 T^{10} - 2572193055285963618 T^{11} - \)\(17\!\cdots\!43\)\( T^{12} - 2572193055285963618 p^{2} T^{13} - 69183145856627091 p^{4} T^{14} - 1186851009182872 p^{6} T^{15} + 10177397826039 p^{8} T^{16} + 583337737242 p^{10} T^{17} + 25757282596 p^{12} T^{18} + 565964145 p^{14} T^{19} + 10863897 p^{16} T^{20} + 294074 p^{18} T^{21} + 6018 p^{20} T^{22} + 99 p^{22} T^{23} + p^{24} T^{24} \) | |
| 53 | \( 1 - 111 T + 13224 T^{2} - 910716 T^{3} + 59831619 T^{4} - 2799852225 T^{5} + 110592030932 T^{6} - 4169273946750 T^{7} + 193735787337603 T^{8} - 18841775739136788 T^{9} + 1586777771876360301 T^{10} - \)\(12\!\cdots\!42\)\( T^{11} + \)\(69\!\cdots\!33\)\( T^{12} - \)\(12\!\cdots\!42\)\( p^{2} T^{13} + 1586777771876360301 p^{4} T^{14} - 18841775739136788 p^{6} T^{15} + 193735787337603 p^{8} T^{16} - 4169273946750 p^{10} T^{17} + 110592030932 p^{12} T^{18} - 2799852225 p^{14} T^{19} + 59831619 p^{16} T^{20} - 910716 p^{18} T^{21} + 13224 p^{20} T^{22} - 111 p^{22} T^{23} + p^{24} T^{24} \) | |
| 59 | \( 1 - 3 T - 5478 T^{2} - 236796 T^{3} + 20289939 T^{4} + 538841139 T^{5} - 70573726772 T^{6} + 365767344396 T^{7} + 479729033250141 T^{8} + 1699477547609316 T^{9} - 1883805256432996599 T^{10} - 14979381777275838012 T^{11} + \)\(59\!\cdots\!37\)\( T^{12} - 14979381777275838012 p^{2} T^{13} - 1883805256432996599 p^{4} T^{14} + 1699477547609316 p^{6} T^{15} + 479729033250141 p^{8} T^{16} + 365767344396 p^{10} T^{17} - 70573726772 p^{12} T^{18} + 538841139 p^{14} T^{19} + 20289939 p^{16} T^{20} - 236796 p^{18} T^{21} - 5478 p^{20} T^{22} - 3 p^{22} T^{23} + p^{24} T^{24} \) | |
| 61 | \( 1 - 150 T + 9552 T^{2} - 165800 T^{3} - 47534514 T^{4} + 4657049670 T^{5} - 161253582314 T^{6} - 6697565984088 T^{7} + 1572100685257950 T^{8} - 92881003919900360 T^{9} + 1478892268641949590 T^{10} + \)\(23\!\cdots\!00\)\( T^{11} - \)\(26\!\cdots\!65\)\( T^{12} + \)\(23\!\cdots\!00\)\( p^{2} T^{13} + 1478892268641949590 p^{4} T^{14} - 92881003919900360 p^{6} T^{15} + 1572100685257950 p^{8} T^{16} - 6697565984088 p^{10} T^{17} - 161253582314 p^{12} T^{18} + 4657049670 p^{14} T^{19} - 47534514 p^{16} T^{20} - 165800 p^{18} T^{21} + 9552 p^{20} T^{22} - 150 p^{22} T^{23} + p^{24} T^{24} \) | |
| 67 | \( 1 - 135 T - 4890 T^{2} + 27630 p T^{3} - 82497135 T^{4} - 6137913249 T^{5} + 840261936376 T^{6} - 28744747210134 T^{7} - 2010008421697905 T^{8} + 265516527753593460 T^{9} - 7719241585001434479 T^{10} - \)\(56\!\cdots\!30\)\( T^{11} + \)\(67\!\cdots\!21\)\( T^{12} - \)\(56\!\cdots\!30\)\( p^{2} T^{13} - 7719241585001434479 p^{4} T^{14} + 265516527753593460 p^{6} T^{15} - 2010008421697905 p^{8} T^{16} - 28744747210134 p^{10} T^{17} + 840261936376 p^{12} T^{18} - 6137913249 p^{14} T^{19} - 82497135 p^{16} T^{20} + 27630 p^{19} T^{21} - 4890 p^{20} T^{22} - 135 p^{22} T^{23} + p^{24} T^{24} \) | |
| 71 | \( 1 + 168 T + 7788 T^{2} - 460248 T^{3} - 78112236 T^{4} - 4942596282 T^{5} - 129339672166 T^{6} + 21694723507134 T^{7} + 3947673961555920 T^{8} + 264008734899691344 T^{9} + 2103086375048613636 T^{10} - \)\(10\!\cdots\!86\)\( T^{11} - \)\(10\!\cdots\!97\)\( T^{12} - \)\(10\!\cdots\!86\)\( p^{2} T^{13} + 2103086375048613636 p^{4} T^{14} + 264008734899691344 p^{6} T^{15} + 3947673961555920 p^{8} T^{16} + 21694723507134 p^{10} T^{17} - 129339672166 p^{12} T^{18} - 4942596282 p^{14} T^{19} - 78112236 p^{16} T^{20} - 460248 p^{18} T^{21} + 7788 p^{20} T^{22} + 168 p^{22} T^{23} + p^{24} T^{24} \) | |
| 73 | \( 1 + 90 T - 1881 T^{2} - 503037 T^{3} - 10955619 T^{4} - 1624145481 T^{5} - 104644983377 T^{6} + 1641371272965 T^{7} - 250831885102848 T^{8} - 82541675177265258 T^{9} + 1186525377744958692 T^{10} + \)\(50\!\cdots\!97\)\( T^{11} + \)\(46\!\cdots\!20\)\( T^{12} + \)\(50\!\cdots\!97\)\( p^{2} T^{13} + 1186525377744958692 p^{4} T^{14} - 82541675177265258 p^{6} T^{15} - 250831885102848 p^{8} T^{16} + 1641371272965 p^{10} T^{17} - 104644983377 p^{12} T^{18} - 1624145481 p^{14} T^{19} - 10955619 p^{16} T^{20} - 503037 p^{18} T^{21} - 1881 p^{20} T^{22} + 90 p^{22} T^{23} + p^{24} T^{24} \) | |
| 79 | \( 1 + 75 T - 2808 T^{2} + 1317084 T^{3} + 113789640 T^{4} - 2453884791 T^{5} + 938263725814 T^{6} + 65842204081428 T^{7} - 934380072746862 T^{8} + 572878760809167420 T^{9} + 18887866671028404426 T^{10} + \)\(18\!\cdots\!16\)\( T^{11} + \)\(30\!\cdots\!83\)\( T^{12} + \)\(18\!\cdots\!16\)\( p^{2} T^{13} + 18887866671028404426 p^{4} T^{14} + 572878760809167420 p^{6} T^{15} - 934380072746862 p^{8} T^{16} + 65842204081428 p^{10} T^{17} + 938263725814 p^{12} T^{18} - 2453884791 p^{14} T^{19} + 113789640 p^{16} T^{20} + 1317084 p^{18} T^{21} - 2808 p^{20} T^{22} + 75 p^{22} T^{23} + p^{24} T^{24} \) | |
| 83 | \( 1 + 156 T - 21894 T^{2} - 3167984 T^{3} + 487243866 T^{4} + 48063809730 T^{5} - 7192617372338 T^{6} - 432106800861834 T^{7} + 89789242349485116 T^{8} + 2881436654419447720 T^{9} - \)\(84\!\cdots\!16\)\( T^{10} - \)\(71\!\cdots\!22\)\( T^{11} + \)\(66\!\cdots\!03\)\( T^{12} - \)\(71\!\cdots\!22\)\( p^{2} T^{13} - \)\(84\!\cdots\!16\)\( p^{4} T^{14} + 2881436654419447720 p^{6} T^{15} + 89789242349485116 p^{8} T^{16} - 432106800861834 p^{10} T^{17} - 7192617372338 p^{12} T^{18} + 48063809730 p^{14} T^{19} + 487243866 p^{16} T^{20} - 3167984 p^{18} T^{21} - 21894 p^{20} T^{22} + 156 p^{22} T^{23} + p^{24} T^{24} \) | |
| 89 | \( 1 + 558 T + 156564 T^{2} + 30167973 T^{3} + 4436202555 T^{4} + 520050206223 T^{5} + 50823330783392 T^{6} + 4459983482310345 T^{7} + 400421887982378415 T^{8} + 41564349081369265482 T^{9} + \)\(47\!\cdots\!03\)\( T^{10} + \)\(51\!\cdots\!21\)\( T^{11} + \)\(49\!\cdots\!65\)\( T^{12} + \)\(51\!\cdots\!21\)\( p^{2} T^{13} + \)\(47\!\cdots\!03\)\( p^{4} T^{14} + 41564349081369265482 p^{6} T^{15} + 400421887982378415 p^{8} T^{16} + 4459983482310345 p^{10} T^{17} + 50823330783392 p^{12} T^{18} + 520050206223 p^{14} T^{19} + 4436202555 p^{16} T^{20} + 30167973 p^{18} T^{21} + 156564 p^{20} T^{22} + 558 p^{22} T^{23} + p^{24} T^{24} \) | |
| 97 | \( 1 - 465 T + 120888 T^{2} - 22632888 T^{3} + 3339144633 T^{4} - 402070083927 T^{5} + 39631627117156 T^{6} - 2930901400124028 T^{7} + 96086821819014021 T^{8} + 15804430080957435372 T^{9} - \)\(40\!\cdots\!67\)\( T^{10} + \)\(57\!\cdots\!16\)\( T^{11} - \)\(62\!\cdots\!95\)\( T^{12} + \)\(57\!\cdots\!16\)\( p^{2} T^{13} - \)\(40\!\cdots\!67\)\( p^{4} T^{14} + 15804430080957435372 p^{6} T^{15} + 96086821819014021 p^{8} T^{16} - 2930901400124028 p^{10} T^{17} + 39631627117156 p^{12} T^{18} - 402070083927 p^{14} T^{19} + 3339144633 p^{16} T^{20} - 22632888 p^{18} T^{21} + 120888 p^{20} T^{22} - 465 p^{22} T^{23} + p^{24} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.85106244590997063393480296330, −7.41168908165541680535077079813, −7.18932761619157437526033486191, −7.08854562074849904480263488623, −7.06774014772897186226511350371, −6.92625471020993493116535833028, −6.51361481316212852347501385985, −6.38341205812640771042556486409, −6.15860181736768261401115256588, −6.06660408216617329579612142416, −5.91534662688780431176772607572, −5.89987384037569998939771384900, −5.81421001133592366861712499621, −5.15118769448373011118286068546, −4.96098224615106101796762514901, −4.75427582003012185626047363568, −4.70263772762855863523261366335, −4.29342940953308004538746109581, −4.09724206788381690037057893811, −4.01832006244486933641761941779, −3.37701672274605221597132550458, −3.36620380436415981056188224225, −2.56909808272184381686094111302, −2.51550721235901519973879287446, −2.16256056132298264774280904277, 2.16256056132298264774280904277, 2.51550721235901519973879287446, 2.56909808272184381686094111302, 3.36620380436415981056188224225, 3.37701672274605221597132550458, 4.01832006244486933641761941779, 4.09724206788381690037057893811, 4.29342940953308004538746109581, 4.70263772762855863523261366335, 4.75427582003012185626047363568, 4.96098224615106101796762514901, 5.15118769448373011118286068546, 5.81421001133592366861712499621, 5.89987384037569998939771384900, 5.91534662688780431176772607572, 6.06660408216617329579612142416, 6.15860181736768261401115256588, 6.38341205812640771042556486409, 6.51361481316212852347501385985, 6.92625471020993493116535833028, 7.06774014772897186226511350371, 7.08854562074849904480263488623, 7.18932761619157437526033486191, 7.41168908165541680535077079813, 7.85106244590997063393480296330
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.