| L(s) = 1 | + 9.09e11·2-s + 7.58e20·3-s − 1.53e26·4-s + 1.79e30·5-s + 6.89e32·6-s − 6.36e36·7-s − 2.80e38·8-s + 2.52e41·9-s + 1.63e42·10-s + 1.26e45·11-s − 1.16e47·12-s − 2.39e48·13-s − 5.78e48·14-s + 1.36e51·15-s + 2.35e52·16-s + 4.72e53·17-s + 2.29e53·18-s + 5.18e55·19-s − 2.76e56·20-s − 4.82e57·21-s + 1.14e57·22-s + 2.83e59·23-s − 2.12e59·24-s − 3.23e60·25-s − 2.18e60·26-s − 5.39e61·27-s + 9.79e62·28-s + ⋯ |
| L(s) = 1 | + 0.0731·2-s + 1.33·3-s − 0.994·4-s + 0.706·5-s + 0.0975·6-s − 1.10·7-s − 0.145·8-s + 0.780·9-s + 0.0516·10-s + 0.631·11-s − 1.32·12-s − 0.838·13-s − 0.0805·14-s + 0.942·15-s + 0.983·16-s + 1.41·17-s + 0.0570·18-s + 1.22·19-s − 0.702·20-s − 1.46·21-s + 0.0461·22-s + 1.64·23-s − 0.194·24-s − 0.501·25-s − 0.0613·26-s − 0.293·27-s + 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(88-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+87/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(2.936499437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.936499437\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 9.09e11T + 1.54e26T^{2} \) |
| 3 | \( 1 - 7.58e20T + 3.23e41T^{2} \) |
| 5 | \( 1 - 1.79e30T + 6.46e60T^{2} \) |
| 7 | \( 1 + 6.36e36T + 3.33e73T^{2} \) |
| 11 | \( 1 - 1.26e45T + 3.99e90T^{2} \) |
| 13 | \( 1 + 2.39e48T + 8.18e96T^{2} \) |
| 17 | \( 1 - 4.72e53T + 1.11e107T^{2} \) |
| 19 | \( 1 - 5.18e55T + 1.78e111T^{2} \) |
| 23 | \( 1 - 2.83e59T + 2.95e118T^{2} \) |
| 29 | \( 1 + 1.35e63T + 1.69e127T^{2} \) |
| 31 | \( 1 + 6.96e64T + 5.60e129T^{2} \) |
| 37 | \( 1 - 2.78e68T + 2.71e136T^{2} \) |
| 41 | \( 1 - 1.17e70T + 2.05e140T^{2} \) |
| 43 | \( 1 - 8.83e70T + 1.29e142T^{2} \) |
| 47 | \( 1 - 4.47e72T + 2.96e145T^{2} \) |
| 53 | \( 1 - 1.55e75T + 1.02e150T^{2} \) |
| 59 | \( 1 + 1.66e77T + 1.15e154T^{2} \) |
| 61 | \( 1 + 6.67e75T + 2.10e155T^{2} \) |
| 67 | \( 1 - 4.39e79T + 7.38e158T^{2} \) |
| 71 | \( 1 - 1.61e80T + 1.14e161T^{2} \) |
| 73 | \( 1 - 1.13e81T + 1.28e162T^{2} \) |
| 79 | \( 1 + 6.04e81T + 1.24e165T^{2} \) |
| 83 | \( 1 - 1.07e83T + 9.11e166T^{2} \) |
| 89 | \( 1 + 4.97e84T + 3.95e169T^{2} \) |
| 97 | \( 1 - 1.77e86T + 7.06e172T^{2} \) |
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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
−14.72992249538577988828595437361, −13.81088972678231426087603457259, −12.69589312964451976875533744860, −9.575687843931762856877189774731, −9.356391353977187961125951112503, −7.55885181619753342918705096543, −5.55722976505826817465571579318, −3.71021238724494027693721385244, −2.75096612516943840057965160499, −0.946401193654180600381204881757,
0.946401193654180600381204881757, 2.75096612516943840057965160499, 3.71021238724494027693721385244, 5.55722976505826817465571579318, 7.55885181619753342918705096543, 9.356391353977187961125951112503, 9.575687843931762856877189774731, 12.69589312964451976875533744860, 13.81088972678231426087603457259, 14.72992249538577988828595437361
