L(s) = 1 | − 2·2-s + 4-s − 7-s − 9-s − 11-s + 2·14-s + 2·18-s + 2·22-s − 2·23-s − 25-s − 28-s − 2·29-s + 2·32-s − 36-s + 3·37-s − 2·43-s − 44-s + 4·46-s + 2·50-s + 3·53-s + 4·58-s + 63-s − 4·64-s − 2·67-s + 3·71-s − 6·74-s + 77-s + ⋯ |
L(s) = 1 | − 2·2-s + 4-s − 7-s − 9-s − 11-s + 2·14-s + 2·18-s + 2·22-s − 2·23-s − 25-s − 28-s − 2·29-s + 2·32-s − 36-s + 3·37-s − 2·43-s − 44-s + 4·46-s + 2·50-s + 3·53-s + 4·58-s + 63-s − 4·64-s − 2·67-s + 3·71-s − 6·74-s + 77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02364688051\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02364688051\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
good | 2 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 29 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14297921968316790348669077014, −10.45975473868778616407928874431, −10.14236648571518692718561215438, −10.12551310524947616653456690813, −9.911879241447849028762294713107, −9.414322799362812733857412722582, −9.212076943517285374974722917022, −9.096411228983060337014424351550, −8.935139995367423919312740922857, −8.145878264529248582217719935070, −8.036248377990937902874141406961, −7.997295040882863506221465232122, −7.86864847338819353586449231672, −7.03454853540565329256094827597, −6.88058601121399683445656354047, −6.14991031457579889100996880776, −5.99237450344020077548331918694, −5.98104918423109067487880098106, −5.16992084082465797227471799327, −5.02305203641943809312723389715, −4.11400874670323536123765252069, −3.89149719521873110298670825170, −3.35405452155751025316205560791, −2.48329704472032446623488429724, −2.35278045266079068339323690453,
2.35278045266079068339323690453, 2.48329704472032446623488429724, 3.35405452155751025316205560791, 3.89149719521873110298670825170, 4.11400874670323536123765252069, 5.02305203641943809312723389715, 5.16992084082465797227471799327, 5.98104918423109067487880098106, 5.99237450344020077548331918694, 6.14991031457579889100996880776, 6.88058601121399683445656354047, 7.03454853540565329256094827597, 7.86864847338819353586449231672, 7.997295040882863506221465232122, 8.036248377990937902874141406961, 8.145878264529248582217719935070, 8.935139995367423919312740922857, 9.096411228983060337014424351550, 9.212076943517285374974722917022, 9.414322799362812733857412722582, 9.911879241447849028762294713107, 10.12551310524947616653456690813, 10.14236648571518692718561215438, 10.45975473868778616407928874431, 11.14297921968316790348669077014