| L(s) = 1 | − 6.77e16·2-s − 4.98e26·3-s − 5.79e33·4-s − 2.60e39·5-s + 3.37e43·6-s + 3.08e47·7-s + 1.09e51·8-s − 5.73e53·9-s + 1.76e56·10-s − 1.29e59·11-s + 2.89e60·12-s − 3.41e62·13-s − 2.08e64·14-s + 1.29e66·15-s − 1.40e67·16-s + 4.55e69·17-s + 3.88e70·18-s − 6.83e71·19-s + 1.51e73·20-s − 1.53e74·21-s + 8.75e75·22-s + 3.21e76·23-s − 5.46e77·24-s − 2.84e78·25-s + 2.31e79·26-s + 6.95e80·27-s − 1.78e81·28-s + ⋯ |
| L(s) = 1 | − 0.664·2-s − 0.550·3-s − 0.558·4-s − 0.839·5-s + 0.365·6-s + 0.550·7-s + 1.03·8-s − 0.697·9-s + 0.557·10-s − 1.87·11-s + 0.307·12-s − 0.393·13-s − 0.365·14-s + 0.461·15-s − 0.129·16-s + 1.37·17-s + 0.463·18-s − 0.384·19-s + 0.468·20-s − 0.302·21-s + 1.24·22-s + 0.370·23-s − 0.569·24-s − 0.295·25-s + 0.261·26-s + 0.933·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(114-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+56.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(57)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{115}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 6.77e16T + 1.03e34T^{2} \) |
| 3 | \( 1 + 4.98e26T + 8.21e53T^{2} \) |
| 5 | \( 1 + 2.60e39T + 9.62e78T^{2} \) |
| 7 | \( 1 - 3.08e47T + 3.13e95T^{2} \) |
| 11 | \( 1 + 1.29e59T + 4.75e117T^{2} \) |
| 13 | \( 1 + 3.41e62T + 7.50e125T^{2} \) |
| 17 | \( 1 - 4.55e69T + 1.09e139T^{2} \) |
| 19 | \( 1 + 6.83e71T + 3.15e144T^{2} \) |
| 23 | \( 1 - 3.21e76T + 7.50e153T^{2} \) |
| 29 | \( 1 - 5.81e82T + 1.78e165T^{2} \) |
| 31 | \( 1 - 8.62e83T + 3.34e168T^{2} \) |
| 37 | \( 1 - 3.36e88T + 1.60e177T^{2} \) |
| 41 | \( 1 - 1.91e91T + 1.75e182T^{2} \) |
| 43 | \( 1 - 9.11e91T + 3.81e184T^{2} \) |
| 47 | \( 1 - 4.23e92T + 8.85e188T^{2} \) |
| 53 | \( 1 + 1.29e97T + 6.96e194T^{2} \) |
| 59 | \( 1 - 8.79e98T + 1.27e200T^{2} \) |
| 61 | \( 1 - 1.49e99T + 5.52e201T^{2} \) |
| 67 | \( 1 - 1.91e103T + 2.22e206T^{2} \) |
| 71 | \( 1 + 4.73e104T + 1.55e209T^{2} \) |
| 73 | \( 1 - 3.08e105T + 3.59e210T^{2} \) |
| 79 | \( 1 + 2.94e107T + 2.70e214T^{2} \) |
| 83 | \( 1 + 3.83e108T + 7.17e216T^{2} \) |
| 89 | \( 1 + 1.59e110T + 1.91e220T^{2} \) |
| 97 | \( 1 + 1.90e112T + 3.20e224T^{2} \) |
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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
−12.62089039068092686640181796402, −11.18170880048869230386603624954, −10.08471899626082956274475020633, −8.279161731050514054288108501151, −7.70112074017409400592788212078, −5.49800641575099339856761403818, −4.54186980639979648952646052581, −2.78635497833156212768402691618, −0.865553418048843576356128939895, 0,
0.865553418048843576356128939895, 2.78635497833156212768402691618, 4.54186980639979648952646052581, 5.49800641575099339856761403818, 7.70112074017409400592788212078, 8.279161731050514054288108501151, 10.08471899626082956274475020633, 11.18170880048869230386603624954, 12.62089039068092686640181796402
