| L(s) = 1 | + 1.52e17·2-s − 1.34e27·3-s + 1.27e34·4-s − 5.29e39·5-s − 2.04e44·6-s + 7.47e47·7-s + 3.58e50·8-s + 9.82e53·9-s − 8.04e56·10-s + 7.05e58·11-s − 1.71e61·12-s + 6.94e62·13-s + 1.13e65·14-s + 7.10e66·15-s − 7.77e67·16-s + 2.71e69·17-s + 1.49e71·18-s − 3.26e71·19-s − 6.74e73·20-s − 1.00e75·21-s + 1.07e76·22-s − 1.19e77·23-s − 4.82e77·24-s + 1.83e79·25-s + 1.05e80·26-s − 2.15e80·27-s + 9.52e81·28-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 1.48·3-s + 1.22·4-s − 1.70·5-s − 2.21·6-s + 1.33·7-s + 0.339·8-s + 1.19·9-s − 2.54·10-s + 1.02·11-s − 1.81·12-s + 0.801·13-s + 1.99·14-s + 2.52·15-s − 0.721·16-s + 0.818·17-s + 1.78·18-s − 0.183·19-s − 2.09·20-s − 1.97·21-s + 1.52·22-s − 1.37·23-s − 0.502·24-s + 1.90·25-s + 1.19·26-s − 0.289·27-s + 1.63·28-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 | \( 1 - 1.52e17T + 1.03e34T^{2} \) |
| 3 | \( 1 + 1.34e27T + 8.21e53T^{2} \) |
| 5 | \( 1 + 5.29e39T + 9.62e78T^{2} \) |
| 7 | \( 1 - 7.47e47T + 3.13e95T^{2} \) |
| 11 | \( 1 - 7.05e58T + 4.75e117T^{2} \) |
| 13 | \( 1 - 6.94e62T + 7.50e125T^{2} \) |
| 17 | \( 1 - 2.71e69T + 1.09e139T^{2} \) |
| 19 | \( 1 + 3.26e71T + 3.15e144T^{2} \) |
| 23 | \( 1 + 1.19e77T + 7.50e153T^{2} \) |
| 29 | \( 1 + 2.89e82T + 1.78e165T^{2} \) |
| 31 | \( 1 + 1.86e82T + 3.34e168T^{2} \) |
| 37 | \( 1 - 2.27e88T + 1.60e177T^{2} \) |
| 41 | \( 1 + 1.45e91T + 1.75e182T^{2} \) |
| 43 | \( 1 + 7.06e91T + 3.81e184T^{2} \) |
| 47 | \( 1 + 1.50e94T + 8.85e188T^{2} \) |
| 53 | \( 1 + 2.21e96T + 6.96e194T^{2} \) |
| 59 | \( 1 + 1.59e100T + 1.27e200T^{2} \) |
| 61 | \( 1 - 7.78e99T + 5.52e201T^{2} \) |
| 67 | \( 1 - 2.32e103T + 2.22e206T^{2} \) |
| 71 | \( 1 - 7.49e104T + 1.55e209T^{2} \) |
| 73 | \( 1 + 1.93e105T + 3.59e210T^{2} \) |
| 79 | \( 1 - 8.06e106T + 2.70e214T^{2} \) |
| 83 | \( 1 + 1.82e108T + 7.17e216T^{2} \) |
| 89 | \( 1 + 1.41e110T + 1.91e220T^{2} \) |
| 97 | \( 1 + 2.96e112T + 3.20e224T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.33277975240151777873269485592, −11.63880846870384482484994993817, −11.12695104065809098543144937444, −8.075974653628504202956599481832, −6.60554455582389978875930554285, −5.38711072014449327405683194175, −4.37336990575956530664521862969, −3.69176413258007089450716703658, −1.35158887820690902201554007166, 0,
1.35158887820690902201554007166, 3.69176413258007089450716703658, 4.37336990575956530664521862969, 5.38711072014449327405683194175, 6.60554455582389978875930554285, 8.075974653628504202956599481832, 11.12695104065809098543144937444, 11.63880846870384482484994993817, 12.33277975240151777873269485592
