| L(s) = 1 | − 4.81e13·2-s − 1.58e21·3-s + 1.69e27·4-s + 1.07e31·5-s + 7.64e34·6-s + 5.70e37·7-s − 5.17e40·8-s − 3.81e41·9-s − 5.18e44·10-s + 1.25e46·11-s − 2.69e48·12-s − 3.04e49·13-s − 2.74e51·14-s − 1.71e52·15-s + 1.44e54·16-s − 1.44e54·17-s + 1.83e55·18-s − 2.63e56·19-s + 1.82e58·20-s − 9.06e58·21-s − 6.05e59·22-s + 1.82e60·23-s + 8.23e61·24-s − 4.53e61·25-s + 1.46e63·26-s + 5.23e63·27-s + 9.66e64·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.932·3-s + 2.73·4-s + 0.848·5-s + 1.80·6-s + 1.41·7-s − 3.36·8-s − 0.131·9-s − 1.63·10-s + 0.572·11-s − 2.55·12-s − 0.819·13-s − 2.72·14-s − 0.790·15-s + 3.76·16-s − 0.253·17-s + 0.253·18-s − 0.328·19-s + 2.32·20-s − 1.31·21-s − 1.10·22-s + 0.462·23-s + 3.13·24-s − 0.280·25-s + 1.58·26-s + 1.05·27-s + 3.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(45)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{91}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 4.81e13T + 6.18e26T^{2} \) |
| 3 | \( 1 + 1.58e21T + 2.90e42T^{2} \) |
| 5 | \( 1 - 1.07e31T + 1.61e62T^{2} \) |
| 7 | \( 1 - 5.70e37T + 1.63e75T^{2} \) |
| 11 | \( 1 - 1.25e46T + 4.83e92T^{2} \) |
| 13 | \( 1 + 3.04e49T + 1.38e99T^{2} \) |
| 17 | \( 1 + 1.44e54T + 3.23e109T^{2} \) |
| 19 | \( 1 + 2.63e56T + 6.44e113T^{2} \) |
| 23 | \( 1 - 1.82e60T + 1.56e121T^{2} \) |
| 29 | \( 1 + 1.87e65T + 1.42e130T^{2} \) |
| 31 | \( 1 - 9.55e65T + 5.38e132T^{2} \) |
| 37 | \( 1 - 6.40e69T + 3.71e139T^{2} \) |
| 41 | \( 1 + 1.04e72T + 3.44e143T^{2} \) |
| 43 | \( 1 + 3.49e72T + 2.39e145T^{2} \) |
| 47 | \( 1 - 6.29e72T + 6.55e148T^{2} \) |
| 53 | \( 1 - 6.67e75T + 2.88e153T^{2} \) |
| 59 | \( 1 + 1.69e78T + 4.03e157T^{2} \) |
| 61 | \( 1 - 1.46e79T + 7.84e158T^{2} \) |
| 67 | \( 1 - 2.58e81T + 3.31e162T^{2} \) |
| 71 | \( 1 - 2.08e82T + 5.78e164T^{2} \) |
| 73 | \( 1 - 2.80e82T + 6.85e165T^{2} \) |
| 79 | \( 1 - 1.45e84T + 7.74e168T^{2} \) |
| 83 | \( 1 + 3.27e85T + 6.27e170T^{2} \) |
| 89 | \( 1 + 3.07e86T + 3.13e173T^{2} \) |
| 97 | \( 1 - 1.70e88T + 6.64e176T^{2} \) |
| 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
−14.84857644081875279050601119220, −11.76975237854152987114441787029, −10.93982418972054598244274876526, −9.600846531089496966527052970740, −8.231197368763280992121787890134, −6.74762281970422833159493819678, −5.40996499257468715029139143785, −2.21905934986599851932169852405, −1.29402578060258513574725849796, 0,
1.29402578060258513574725849796, 2.21905934986599851932169852405, 5.40996499257468715029139143785, 6.74762281970422833159493819678, 8.231197368763280992121787890134, 9.600846531089496966527052970740, 10.93982418972054598244274876526, 11.76975237854152987114441787029, 14.84857644081875279050601119220
