| L(s) = 1 | − 1.88e17·2-s + 4.27e26·3-s + 2.51e34·4-s − 5.60e39·5-s − 8.05e43·6-s − 6.14e47·7-s − 2.79e51·8-s − 6.39e53·9-s + 1.05e57·10-s + 8.11e58·11-s + 1.07e61·12-s − 4.04e62·13-s + 1.15e65·14-s − 2.39e66·15-s + 2.65e68·16-s − 4.66e68·17-s + 1.20e71·18-s + 7.05e71·19-s − 1.41e74·20-s − 2.62e74·21-s − 1.53e76·22-s + 7.73e75·23-s − 1.19e78·24-s + 2.17e79·25-s + 7.62e79·26-s − 6.23e80·27-s − 1.54e82·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.471·3-s + 2.42·4-s − 1.80·5-s − 0.871·6-s − 1.09·7-s − 2.63·8-s − 0.778·9-s + 3.34·10-s + 1.17·11-s + 1.14·12-s − 0.466·13-s + 2.03·14-s − 0.850·15-s + 2.45·16-s − 0.140·17-s + 1.44·18-s + 0.397·19-s − 4.37·20-s − 0.516·21-s − 2.17·22-s + 0.0892·23-s − 1.24·24-s + 2.25·25-s + 0.863·26-s − 0.837·27-s − 2.66·28-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 | \( 1 + 1.88e17T + 1.03e34T^{2} \) |
| 3 | \( 1 - 4.27e26T + 8.21e53T^{2} \) |
| 5 | \( 1 + 5.60e39T + 9.62e78T^{2} \) |
| 7 | \( 1 + 6.14e47T + 3.13e95T^{2} \) |
| 11 | \( 1 - 8.11e58T + 4.75e117T^{2} \) |
| 13 | \( 1 + 4.04e62T + 7.50e125T^{2} \) |
| 17 | \( 1 + 4.66e68T + 1.09e139T^{2} \) |
| 19 | \( 1 - 7.05e71T + 3.15e144T^{2} \) |
| 23 | \( 1 - 7.73e75T + 7.50e153T^{2} \) |
| 29 | \( 1 + 7.64e81T + 1.78e165T^{2} \) |
| 31 | \( 1 - 8.29e83T + 3.34e168T^{2} \) |
| 37 | \( 1 - 4.92e88T + 1.60e177T^{2} \) |
| 41 | \( 1 + 1.22e91T + 1.75e182T^{2} \) |
| 43 | \( 1 - 2.60e92T + 3.81e184T^{2} \) |
| 47 | \( 1 + 1.88e94T + 8.85e188T^{2} \) |
| 53 | \( 1 - 3.65e97T + 6.96e194T^{2} \) |
| 59 | \( 1 - 1.85e99T + 1.27e200T^{2} \) |
| 61 | \( 1 + 7.04e100T + 5.52e201T^{2} \) |
| 67 | \( 1 - 2.94e103T + 2.22e206T^{2} \) |
| 71 | \( 1 + 1.56e104T + 1.55e209T^{2} \) |
| 73 | \( 1 - 2.45e105T + 3.59e210T^{2} \) |
| 79 | \( 1 + 1.45e107T + 2.70e214T^{2} \) |
| 83 | \( 1 + 2.11e106T + 7.17e216T^{2} \) |
| 89 | \( 1 - 1.20e110T + 1.91e220T^{2} \) |
| 97 | \( 1 - 5.43e111T + 3.20e224T^{2} \) |
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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
−12.08497659919003547194686341340, −11.22593667981058123308654946921, −9.562507514488202601559168907827, −8.611368347491042218969088045154, −7.60068946923780987289348158409, −6.56566209504108180314202752331, −3.69782146399026998383265608981, −2.68639789712171710787605528362, −0.847374379631313661895627818077, 0,
0.847374379631313661895627818077, 2.68639789712171710787605528362, 3.69782146399026998383265608981, 6.56566209504108180314202752331, 7.60068946923780987289348158409, 8.611368347491042218969088045154, 9.562507514488202601559168907827, 11.22593667981058123308654946921, 12.08497659919003547194686341340
