| L(s) = 1 | − 7.31e14·2-s − 2.11e23·3-s − 9.86e28·4-s + 5.46e34·5-s + 1.54e38·6-s − 9.42e41·7-s + 5.35e44·8-s − 1.27e47·9-s − 3.99e49·10-s + 6.80e51·11-s + 2.08e52·12-s + 1.27e55·13-s + 6.89e56·14-s − 1.15e58·15-s − 3.29e59·16-s + 1.40e60·17-s + 9.29e61·18-s − 1.83e63·19-s − 5.38e63·20-s + 1.99e65·21-s − 4.98e66·22-s + 7.80e66·23-s − 1.13e68·24-s + 1.40e69·25-s − 9.30e69·26-s + 6.31e70·27-s + 9.29e70·28-s + ⋯ |
| L(s) = 1 | − 0.918·2-s − 0.510·3-s − 0.155·4-s + 1.37·5-s + 0.468·6-s − 1.38·7-s + 1.06·8-s − 0.739·9-s − 1.26·10-s + 1.92·11-s + 0.0794·12-s + 0.921·13-s + 1.27·14-s − 0.701·15-s − 0.820·16-s + 0.174·17-s + 0.679·18-s − 0.920·19-s − 0.214·20-s + 0.707·21-s − 1.76·22-s + 0.306·23-s − 0.541·24-s + 0.890·25-s − 0.846·26-s + 0.887·27-s + 0.215·28-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
| \(L(50)\) |
\(\approx\) |
\(0.9548384322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9548384322\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 7.31e14T + 6.33e29T^{2} \) |
| 3 | \( 1 + 2.11e23T + 1.71e47T^{2} \) |
| 5 | \( 1 - 5.46e34T + 1.57e69T^{2} \) |
| 7 | \( 1 + 9.42e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 6.80e51T + 1.25e103T^{2} \) |
| 13 | \( 1 - 1.27e55T + 1.90e110T^{2} \) |
| 17 | \( 1 - 1.40e60T + 6.52e121T^{2} \) |
| 19 | \( 1 + 1.83e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 7.80e66T + 6.47e134T^{2} \) |
| 29 | \( 1 + 4.49e72T + 5.98e144T^{2} \) |
| 31 | \( 1 + 6.78e73T + 4.41e147T^{2} \) |
| 37 | \( 1 - 1.72e77T + 1.78e155T^{2} \) |
| 41 | \( 1 + 3.13e78T + 4.63e159T^{2} \) |
| 43 | \( 1 - 1.67e80T + 5.16e161T^{2} \) |
| 47 | \( 1 - 5.53e82T + 3.44e165T^{2} \) |
| 53 | \( 1 + 9.83e84T + 5.05e170T^{2} \) |
| 59 | \( 1 + 2.85e87T + 2.06e175T^{2} \) |
| 61 | \( 1 + 7.36e87T + 5.59e176T^{2} \) |
| 67 | \( 1 - 2.22e90T + 6.04e180T^{2} \) |
| 71 | \( 1 - 7.03e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 1.65e92T + 2.94e184T^{2} \) |
| 79 | \( 1 + 2.18e93T + 7.32e187T^{2} \) |
| 83 | \( 1 + 6.95e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 8.71e95T + 9.76e192T^{2} \) |
| 97 | \( 1 - 1.85e98T + 4.90e196T^{2} \) |
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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
−14.16799565798148997783595569686, −12.93833720660445183875023419923, −10.94469032765473633436760393852, −9.503846208621550766386133150816, −8.990317290444552722219284196692, −6.60895979457736553330868038347, −5.79436402776081176624794938184, −3.72735732523573372038775861610, −1.80804204372360563700184258015, −0.63471210310869948932054908978,
0.63471210310869948932054908978, 1.80804204372360563700184258015, 3.72735732523573372038775861610, 5.79436402776081176624794938184, 6.60895979457736553330868038347, 8.990317290444552722219284196692, 9.503846208621550766386133150816, 10.94469032765473633436760393852, 12.93833720660445183875023419923, 14.16799565798148997783595569686
