L(s) = 1 | + (−1 − 2.64i)2-s + 5.29i·3-s + (−6.00 + 5.29i)4-s − 10.5i·5-s + (14.0 − 5.29i)6-s − 8·7-s + (20.0 + 10.5i)8-s − 1.00·9-s + (−28.0 + 10.5i)10-s − 15.8i·11-s + (−28.0 − 31.7i)12-s + 52.9i·13-s + (8 + 21.1i)14-s + 56.0·15-s + (8.00 − 63.4i)16-s − 14·17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.935i)2-s + 1.01i·3-s + (−0.750 + 0.661i)4-s − 0.946i·5-s + (0.952 − 0.360i)6-s − 0.431·7-s + (0.883 + 0.467i)8-s − 0.0370·9-s + (−0.885 + 0.334i)10-s − 0.435i·11-s + (−0.673 − 0.763i)12-s + 1.12i·13-s + (0.152 + 0.404i)14-s + 0.963·15-s + (0.125 − 0.992i)16-s − 0.199·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.644021 - 0.159889i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.644021 - 0.159889i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 2.64i)T \) |
good | 3 | \( 1 - 5.29iT - 27T^{2} \) |
| 5 | \( 1 + 10.5iT - 125T^{2} \) |
| 7 | \( 1 + 8T + 343T^{2} \) |
| 11 | \( 1 + 15.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 52.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 14T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 152T + 1.21e4T^{2} \) |
| 29 | \( 1 + 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 + 439. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 336T + 1.03e5T^{2} \) |
| 53 | \( 1 - 31.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 534. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 95.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 174. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 72T + 3.57e5T^{2} \) |
| 73 | \( 1 + 294T + 3.89e5T^{2} \) |
| 79 | \( 1 + 464T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.30376301435529161239728484600, −20.32217722439891987038930398912, −18.94139912399034712115163287588, −17.01365951247526908992298118503, −15.95486277328282696300256864193, −13.56118051764597305259579034934, −11.87562116271811096424835068025, −10.06406893318100641723738699310, −8.832485509020613984625872254045, −4.35379543989648781937842455545,
6.41965094825064143108164280420, 7.76460757569393509809552959748, 10.19856353228435028276780554201, 12.80093509323794710250940362899, 14.30533482400885281017161563303, 15.76870394334660738848907925024, 17.71952466704451273167168990255, 18.47465037908060946616208160159, 19.68782470900856199067021394123, 22.32328609141571665667068374535