| L(s) = 1 | + 2.86e14·2-s + 2.01e23·3-s − 5.51e29·4-s − 4.79e34·5-s + 5.77e37·6-s − 6.92e41·7-s − 3.39e44·8-s − 1.31e47·9-s − 1.37e49·10-s + 1.55e51·11-s − 1.11e53·12-s − 2.14e55·13-s − 1.98e56·14-s − 9.67e57·15-s + 2.52e59·16-s − 1.39e61·17-s − 3.75e61·18-s + 1.51e63·19-s + 2.64e64·20-s − 1.39e65·21-s + 4.45e65·22-s + 2.55e67·23-s − 6.84e67·24-s + 7.25e68·25-s − 6.14e69·26-s − 6.10e70·27-s + 3.82e71·28-s + ⋯ |
| L(s) = 1 | + 0.359·2-s + 0.486·3-s − 0.870·4-s − 1.20·5-s + 0.174·6-s − 1.01·7-s − 0.673·8-s − 0.763·9-s − 0.434·10-s + 0.439·11-s − 0.423·12-s − 1.55·13-s − 0.366·14-s − 0.587·15-s + 0.628·16-s − 1.72·17-s − 0.274·18-s + 0.762·19-s + 1.05·20-s − 0.495·21-s + 0.158·22-s + 1.00·23-s − 0.327·24-s + 0.459·25-s − 0.559·26-s − 0.857·27-s + 0.886·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(100-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+99/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(50)\) |
\(\approx\) |
\(0.3222349486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3222349486\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.86e14T + 6.33e29T^{2} \) |
| 3 | \( 1 - 2.01e23T + 1.71e47T^{2} \) |
| 5 | \( 1 + 4.79e34T + 1.57e69T^{2} \) |
| 7 | \( 1 + 6.92e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 1.55e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 2.14e55T + 1.90e110T^{2} \) |
| 17 | \( 1 + 1.39e61T + 6.52e121T^{2} \) |
| 19 | \( 1 - 1.51e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 2.55e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 1.39e72T + 5.98e144T^{2} \) |
| 31 | \( 1 + 1.44e73T + 4.41e147T^{2} \) |
| 37 | \( 1 + 5.19e77T + 1.78e155T^{2} \) |
| 41 | \( 1 + 3.58e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 2.84e80T + 5.16e161T^{2} \) |
| 47 | \( 1 + 7.73e82T + 3.44e165T^{2} \) |
| 53 | \( 1 - 1.89e84T + 5.05e170T^{2} \) |
| 59 | \( 1 + 2.13e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 4.67e88T + 5.59e176T^{2} \) |
| 67 | \( 1 - 3.33e90T + 6.04e180T^{2} \) |
| 71 | \( 1 - 2.25e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 2.47e90T + 2.94e184T^{2} \) |
| 79 | \( 1 + 1.29e94T + 7.32e187T^{2} \) |
| 83 | \( 1 + 1.54e95T + 9.74e189T^{2} \) |
| 89 | \( 1 - 2.29e96T + 9.76e192T^{2} \) |
| 97 | \( 1 + 3.70e98T + 4.90e196T^{2} \) |
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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
−14.35657069944626977501733975689, −12.94114628893825834714172890382, −11.68129292927046365872580284471, −9.538678245021351101338331327905, −8.486321174388953504646530580615, −6.92080710886558857504747734174, −4.99422428996055355061406310952, −3.75410456219660588955378331928, −2.79287991358273355125780156377, −0.26531802737573460571075720343,
0.26531802737573460571075720343, 2.79287991358273355125780156377, 3.75410456219660588955378331928, 4.99422428996055355061406310952, 6.92080710886558857504747734174, 8.486321174388953504646530580615, 9.538678245021351101338331327905, 11.68129292927046365872580284471, 12.94114628893825834714172890382, 14.35657069944626977501733975689
