| L(s) = 1 | − 2.38e17·2-s + 4.64e27·3-s + 1.55e34·4-s − 1.89e40·5-s − 1.10e45·6-s + 5.62e48·7-s + 6.21e51·8-s + 1.42e55·9-s + 4.51e57·10-s + 3.76e59·11-s + 7.20e61·12-s + 1.05e64·13-s − 1.34e66·14-s − 8.78e67·15-s − 2.12e69·16-s + 8.69e70·17-s − 3.39e72·18-s − 1.20e72·19-s − 2.93e74·20-s + 2.61e76·21-s − 8.99e76·22-s − 1.80e78·23-s + 2.88e79·24-s + 1.16e80·25-s − 2.52e81·26-s + 3.16e82·27-s + 8.72e82·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 1.70·3-s + 0.373·4-s − 1.21·5-s − 2.00·6-s + 1.43·7-s + 0.734·8-s + 1.92·9-s + 1.42·10-s + 0.496·11-s + 0.637·12-s + 0.937·13-s − 1.68·14-s − 2.08·15-s − 1.23·16-s + 1.54·17-s − 2.25·18-s − 0.0356·19-s − 0.454·20-s + 2.45·21-s − 0.581·22-s − 0.903·23-s + 1.25·24-s + 0.484·25-s − 1.09·26-s + 1.57·27-s + 0.535·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(116-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(58)\) |
\(\approx\) |
\(2.440845840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.440845840\) |
| \(L(\frac{117}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.38e17T + 4.15e34T^{2} \) |
| 3 | \( 1 - 4.64e27T + 7.39e54T^{2} \) |
| 5 | \( 1 + 1.89e40T + 2.40e80T^{2} \) |
| 7 | \( 1 - 5.62e48T + 1.53e97T^{2} \) |
| 11 | \( 1 - 3.76e59T + 5.75e119T^{2} \) |
| 13 | \( 1 - 1.05e64T + 1.26e128T^{2} \) |
| 17 | \( 1 - 8.69e70T + 3.17e141T^{2} \) |
| 19 | \( 1 + 1.20e72T + 1.13e147T^{2} \) |
| 23 | \( 1 + 1.80e78T + 3.96e156T^{2} \) |
| 29 | \( 1 - 1.92e84T + 1.49e168T^{2} \) |
| 31 | \( 1 + 1.96e85T + 3.21e171T^{2} \) |
| 37 | \( 1 - 3.46e88T + 2.20e180T^{2} \) |
| 41 | \( 1 + 5.49e92T + 2.95e185T^{2} \) |
| 43 | \( 1 - 3.36e93T + 7.06e187T^{2} \) |
| 47 | \( 1 + 1.92e96T + 1.95e192T^{2} \) |
| 53 | \( 1 - 2.07e99T + 1.95e198T^{2} \) |
| 59 | \( 1 - 6.48e101T + 4.44e203T^{2} \) |
| 61 | \( 1 + 2.91e102T + 2.05e205T^{2} \) |
| 67 | \( 1 + 7.20e104T + 9.96e209T^{2} \) |
| 71 | \( 1 + 5.41e105T + 7.84e212T^{2} \) |
| 73 | \( 1 + 1.93e107T + 1.91e214T^{2} \) |
| 79 | \( 1 + 2.35e108T + 1.68e218T^{2} \) |
| 83 | \( 1 - 2.87e110T + 4.94e220T^{2} \) |
| 89 | \( 1 - 6.53e111T + 1.51e224T^{2} \) |
| 97 | \( 1 + 1.19e114T + 3.01e228T^{2} \) |
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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
−13.85687972211435742799589017940, −11.73930533013468658469753747012, −10.18226024846304458446103166606, −8.656694769982556812407891223044, −8.175252132881209867892786259417, −7.46521630730476169197110776938, −4.43559469851372251035128747641, −3.46603929778179855779797174640, −1.78718175654514626265241952217, −0.957818787664903702988713135089,
0.957818787664903702988713135089, 1.78718175654514626265241952217, 3.46603929778179855779797174640, 4.43559469851372251035128747641, 7.46521630730476169197110776938, 8.175252132881209867892786259417, 8.656694769982556812407891223044, 10.18226024846304458446103166606, 11.73930533013468658469753747012, 13.85687972211435742799589017940
