| L(s) = 1 | − 6.35e14·2-s + 5.52e23·3-s − 2.30e29·4-s + 4.78e33·5-s − 3.51e38·6-s + 7.17e41·7-s + 5.48e44·8-s + 1.33e47·9-s − 3.03e48·10-s − 4.28e51·11-s − 1.27e53·12-s + 1.83e55·13-s − 4.55e56·14-s + 2.64e57·15-s − 2.02e59·16-s + 6.21e59·17-s − 8.49e61·18-s + 2.46e63·19-s − 1.10e63·20-s + 3.96e65·21-s + 2.71e66·22-s + 1.40e67·23-s + 3.03e68·24-s − 1.55e69·25-s − 1.16e70·26-s − 2.10e70·27-s − 1.65e71·28-s + ⋯ |
| L(s) = 1 | − 0.797·2-s + 1.33·3-s − 0.363·4-s + 0.120·5-s − 1.06·6-s + 1.05·7-s + 1.08·8-s + 0.778·9-s − 0.0960·10-s − 1.20·11-s − 0.484·12-s + 1.33·13-s − 0.842·14-s + 0.160·15-s − 0.504·16-s + 0.0769·17-s − 0.621·18-s + 1.24·19-s − 0.0437·20-s + 1.40·21-s + 0.964·22-s + 0.553·23-s + 1.45·24-s − 0.985·25-s − 1.06·26-s − 0.294·27-s − 0.383·28-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
| \(L(50)\) |
\(\approx\) |
\(2.393977829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393977829\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 6.35e14T + 6.33e29T^{2} \) |
| 3 | \( 1 - 5.52e23T + 1.71e47T^{2} \) |
| 5 | \( 1 - 4.78e33T + 1.57e69T^{2} \) |
| 7 | \( 1 - 7.17e41T + 4.62e83T^{2} \) |
| 11 | \( 1 + 4.28e51T + 1.25e103T^{2} \) |
| 13 | \( 1 - 1.83e55T + 1.90e110T^{2} \) |
| 17 | \( 1 - 6.21e59T + 6.52e121T^{2} \) |
| 19 | \( 1 - 2.46e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 1.40e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 2.85e72T + 5.98e144T^{2} \) |
| 31 | \( 1 + 7.00e73T + 4.41e147T^{2} \) |
| 37 | \( 1 + 5.39e76T + 1.78e155T^{2} \) |
| 41 | \( 1 - 1.04e80T + 4.63e159T^{2} \) |
| 43 | \( 1 + 1.06e81T + 5.16e161T^{2} \) |
| 47 | \( 1 - 6.71e82T + 3.44e165T^{2} \) |
| 53 | \( 1 + 2.07e85T + 5.05e170T^{2} \) |
| 59 | \( 1 + 1.47e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 2.22e88T + 5.59e176T^{2} \) |
| 67 | \( 1 - 4.24e90T + 6.04e180T^{2} \) |
| 71 | \( 1 - 4.39e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 1.41e92T + 2.94e184T^{2} \) |
| 79 | \( 1 - 1.10e94T + 7.32e187T^{2} \) |
| 83 | \( 1 - 8.17e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 5.46e96T + 9.76e192T^{2} \) |
| 97 | \( 1 - 1.47e98T + 4.90e196T^{2} \) |
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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
−14.18822175072361188757243281545, −13.36018533487627125253021599809, −10.91371087954203402749245013439, −9.460541654247393444811221689019, −8.317758829140668945961517233193, −7.73244084943458125999885036766, −5.14762980383696484075228466109, −3.58671410904994897755381279588, −2.08859696690179104973123944383, −0.933902180534301809992712401331,
0.933902180534301809992712401331, 2.08859696690179104973123944383, 3.58671410904994897755381279588, 5.14762980383696484075228466109, 7.73244084943458125999885036766, 8.317758829140668945961517233193, 9.460541654247393444811221689019, 10.91371087954203402749245013439, 13.36018533487627125253021599809, 14.18822175072361188757243281545
