| L(s) = 1 | − 3.55e17·2-s − 3.54e26·3-s + 8.48e34·4-s + 1.00e40·5-s + 1.26e44·6-s + 6.58e47·7-s − 1.54e52·8-s − 7.26e54·9-s − 3.57e57·10-s − 1.00e60·11-s − 3.01e61·12-s + 1.22e64·13-s − 2.34e65·14-s − 3.57e66·15-s + 1.95e69·16-s − 4.28e70·17-s + 2.58e72·18-s − 1.26e73·19-s + 8.54e74·20-s − 2.33e74·21-s + 3.57e77·22-s + 3.82e78·23-s + 5.46e78·24-s − 1.39e80·25-s − 4.36e81·26-s + 5.20e81·27-s + 5.59e82·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.130·3-s + 2.04·4-s + 0.648·5-s + 0.227·6-s + 0.168·7-s − 1.81·8-s − 0.982·9-s − 1.13·10-s − 1.32·11-s − 0.266·12-s + 1.08·13-s − 0.293·14-s − 0.0846·15-s + 1.13·16-s − 0.761·17-s + 1.71·18-s − 0.373·19-s + 1.32·20-s − 0.0219·21-s + 2.31·22-s + 1.92·23-s + 0.237·24-s − 0.579·25-s − 1.90·26-s + 0.258·27-s + 0.343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(116-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(58)\) |
\(\approx\) |
\(0.6146916196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6146916196\) |
| \(L(\frac{117}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 3.55e17T + 4.15e34T^{2} \) |
| 3 | \( 1 + 3.54e26T + 7.39e54T^{2} \) |
| 5 | \( 1 - 1.00e40T + 2.40e80T^{2} \) |
| 7 | \( 1 - 6.58e47T + 1.53e97T^{2} \) |
| 11 | \( 1 + 1.00e60T + 5.75e119T^{2} \) |
| 13 | \( 1 - 1.22e64T + 1.26e128T^{2} \) |
| 17 | \( 1 + 4.28e70T + 3.17e141T^{2} \) |
| 19 | \( 1 + 1.26e73T + 1.13e147T^{2} \) |
| 23 | \( 1 - 3.82e78T + 3.96e156T^{2} \) |
| 29 | \( 1 - 1.01e84T + 1.49e168T^{2} \) |
| 31 | \( 1 + 8.37e85T + 3.21e171T^{2} \) |
| 37 | \( 1 - 5.67e89T + 2.20e180T^{2} \) |
| 41 | \( 1 + 9.82e92T + 2.95e185T^{2} \) |
| 43 | \( 1 + 3.42e93T + 7.06e187T^{2} \) |
| 47 | \( 1 - 1.13e96T + 1.95e192T^{2} \) |
| 53 | \( 1 - 2.14e99T + 1.95e198T^{2} \) |
| 59 | \( 1 + 7.54e101T + 4.44e203T^{2} \) |
| 61 | \( 1 - 4.09e102T + 2.05e205T^{2} \) |
| 67 | \( 1 + 4.09e104T + 9.96e209T^{2} \) |
| 71 | \( 1 - 3.81e106T + 7.84e212T^{2} \) |
| 73 | \( 1 - 9.17e105T + 1.91e214T^{2} \) |
| 79 | \( 1 + 5.88e108T + 1.68e218T^{2} \) |
| 83 | \( 1 + 1.37e110T + 4.94e220T^{2} \) |
| 89 | \( 1 - 4.88e111T + 1.51e224T^{2} \) |
| 97 | \( 1 + 1.09e114T + 3.01e228T^{2} \) |
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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
−13.35048623599355220772292475387, −11.24844841604299967779253588295, −10.50370182887857520103314379705, −9.041786479399691277843689258779, −8.224590510978293394892275398377, −6.73771077690557813259468695429, −5.41986915283946528048728887798, −2.86264795741953416769279116740, −1.79397585629356614004371647850, −0.50510221753417860837524768044,
0.50510221753417860837524768044, 1.79397585629356614004371647850, 2.86264795741953416769279116740, 5.41986915283946528048728887798, 6.73771077690557813259468695429, 8.224590510978293394892275398377, 9.041786479399691277843689258779, 10.50370182887857520103314379705, 11.24844841604299967779253588295, 13.35048623599355220772292475387
