| L(s) = 1 | + 5.65e17·2-s + 1.40e28·3-s + 1.53e35·4-s − 1.09e41·5-s + 7.93e45·6-s + 1.38e48·7-s − 7.13e51·8-s + 1.30e56·9-s − 6.20e58·10-s − 6.70e60·11-s + 2.15e63·12-s − 2.52e65·13-s + 7.84e65·14-s − 1.53e69·15-s − 2.95e70·16-s + 7.77e71·17-s + 7.37e73·18-s + 4.14e74·19-s − 1.68e76·20-s + 1.94e76·21-s − 3.78e78·22-s − 4.89e79·23-s − 1.00e80·24-s + 6.01e81·25-s − 1.42e83·26-s + 8.97e83·27-s + 2.13e83·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 1.72·3-s + 0.924·4-s − 1.41·5-s + 2.38·6-s + 0.0505·7-s − 0.105·8-s + 1.96·9-s − 1.96·10-s − 0.802·11-s + 1.58·12-s − 1.72·13-s + 0.0701·14-s − 2.43·15-s − 1.07·16-s + 0.812·17-s + 2.71·18-s + 0.646·19-s − 1.30·20-s + 0.0870·21-s − 1.11·22-s − 1.06·23-s − 0.181·24-s + 0.998·25-s − 2.39·26-s + 1.65·27-s + 0.0467·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(118-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+58.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(59)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{119}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 5.65e17T + 1.66e35T^{2} \) |
| 3 | \( 1 - 1.40e28T + 6.65e55T^{2} \) |
| 5 | \( 1 + 1.09e41T + 6.01e81T^{2} \) |
| 7 | \( 1 - 1.38e48T + 7.52e98T^{2} \) |
| 11 | \( 1 + 6.70e60T + 6.96e121T^{2} \) |
| 13 | \( 1 + 2.52e65T + 2.14e130T^{2} \) |
| 17 | \( 1 - 7.77e71T + 9.17e143T^{2} \) |
| 19 | \( 1 - 4.14e74T + 4.11e149T^{2} \) |
| 23 | \( 1 + 4.89e79T + 2.09e159T^{2} \) |
| 29 | \( 1 + 6.48e85T + 1.26e171T^{2} \) |
| 31 | \( 1 - 1.40e87T + 3.08e174T^{2} \) |
| 37 | \( 1 - 2.00e89T + 3.01e183T^{2} \) |
| 41 | \( 1 + 2.38e94T + 4.96e188T^{2} \) |
| 43 | \( 1 - 3.49e95T + 1.30e191T^{2} \) |
| 47 | \( 1 + 1.40e97T + 4.31e195T^{2} \) |
| 53 | \( 1 + 2.80e100T + 5.49e201T^{2} \) |
| 59 | \( 1 + 1.78e103T + 1.54e207T^{2} \) |
| 61 | \( 1 - 3.95e104T + 7.64e208T^{2} \) |
| 67 | \( 1 - 6.50e106T + 4.47e213T^{2} \) |
| 71 | \( 1 + 1.52e108T + 3.95e216T^{2} \) |
| 73 | \( 1 - 9.50e108T + 1.02e218T^{2} \) |
| 79 | \( 1 + 3.30e110T + 1.05e222T^{2} \) |
| 83 | \( 1 - 5.76e111T + 3.40e224T^{2} \) |
| 89 | \( 1 - 5.94e113T + 1.19e228T^{2} \) |
| 97 | \( 1 + 2.25e115T + 2.83e232T^{2} \) |
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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
−12.88401199775646865041632087502, −11.87358909472150402469255944461, −9.680538728645778722360355968313, −8.025889500964732236776582571971, −7.35713701196683092277174872686, −5.05044274372146753174859321569, −3.94413044967518093813253795304, −3.18364495187382794127371731700, −2.24417974729927893389916508838, 0,
2.24417974729927893389916508838, 3.18364495187382794127371731700, 3.94413044967518093813253795304, 5.05044274372146753174859321569, 7.35713701196683092277174872686, 8.025889500964732236776582571971, 9.680538728645778722360355968313, 11.87358909472150402469255944461, 12.88401199775646865041632087502
