| L(s) = 1 | − 2.69e17·2-s + 3.14e28·3-s − 2.58e36·4-s − 3.76e42·5-s − 8.45e45·6-s + 1.39e51·7-s + 1.41e54·8-s − 4.40e57·9-s + 1.01e60·10-s + 6.35e62·11-s − 8.12e64·12-s − 2.58e67·13-s − 3.74e68·14-s − 1.18e71·15-s + 6.49e72·16-s + 1.31e74·17-s + 1.18e75·18-s + 3.91e77·19-s + 9.73e78·20-s + 4.37e79·21-s − 1.71e80·22-s − 1.47e82·23-s + 4.43e82·24-s + 1.03e85·25-s + 6.96e84·26-s − 3.07e86·27-s − 3.59e87·28-s + ⋯ |
| L(s) = 1 | − 0.165·2-s + 0.427·3-s − 0.972·4-s − 1.94·5-s − 0.0705·6-s + 1.03·7-s + 0.325·8-s − 0.817·9-s + 0.320·10-s + 0.629·11-s − 0.416·12-s − 1.04·13-s − 0.170·14-s − 0.829·15-s + 0.919·16-s + 0.474·17-s + 0.134·18-s + 1.69·19-s + 1.88·20-s + 0.442·21-s − 0.103·22-s − 0.607·23-s + 0.139·24-s + 2.76·25-s + 0.172·26-s − 0.777·27-s − 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(122-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+60.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(61)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{123}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.69e17T + 2.65e36T^{2} \) |
| 3 | \( 1 - 3.14e28T + 5.39e57T^{2} \) |
| 5 | \( 1 + 3.76e42T + 3.76e84T^{2} \) |
| 7 | \( 1 - 1.39e51T + 1.80e102T^{2} \) |
| 11 | \( 1 - 6.35e62T + 1.01e126T^{2} \) |
| 13 | \( 1 + 2.58e67T + 6.12e134T^{2} \) |
| 17 | \( 1 - 1.31e74T + 7.66e148T^{2} \) |
| 19 | \( 1 - 3.91e77T + 5.36e154T^{2} \) |
| 23 | \( 1 + 1.47e82T + 5.87e164T^{2} \) |
| 29 | \( 1 - 5.75e87T + 8.91e176T^{2} \) |
| 31 | \( 1 + 5.74e89T + 2.84e180T^{2} \) |
| 37 | \( 1 - 1.13e95T + 5.65e189T^{2} \) |
| 41 | \( 1 - 4.70e97T + 1.40e195T^{2} \) |
| 43 | \( 1 + 5.86e98T + 4.46e197T^{2} \) |
| 47 | \( 1 + 6.97e100T + 2.10e202T^{2} \) |
| 53 | \( 1 + 4.13e103T + 4.33e208T^{2} \) |
| 59 | \( 1 + 2.11e107T + 1.87e214T^{2} \) |
| 61 | \( 1 - 7.77e107T + 1.05e216T^{2} \) |
| 67 | \( 1 + 2.02e110T + 9.01e220T^{2} \) |
| 71 | \( 1 - 5.80e111T + 1.00e224T^{2} \) |
| 73 | \( 1 + 5.23e111T + 2.89e225T^{2} \) |
| 79 | \( 1 - 5.52e114T + 4.10e229T^{2} \) |
| 83 | \( 1 - 1.23e116T + 1.61e232T^{2} \) |
| 89 | \( 1 + 1.22e118T + 7.51e235T^{2} \) |
| 97 | \( 1 + 6.18e119T + 2.50e240T^{2} \) |
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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
−12.17059905447872326611445077014, −11.33884504010850561500680403101, −9.361157176897459580039053021145, −8.073364454097412694355146005019, −7.66189225974059914139305130637, −5.11080148755497392592850986875, −4.14065537164408417517156886341, −3.09857052344960047162551893851, −1.04302590719381750002896100873, 0,
1.04302590719381750002896100873, 3.09857052344960047162551893851, 4.14065537164408417517156886341, 5.11080148755497392592850986875, 7.66189225974059914139305130637, 8.073364454097412694355146005019, 9.361157176897459580039053021145, 11.33884504010850561500680403101, 12.17059905447872326611445077014
