| L(s) = 1 | − 4.19e6·2-s + 3.80e10·3-s + 1.75e13·4-s + 8.48e14·5-s − 1.59e17·6-s − 4.24e18·7-s − 7.37e19·8-s − 1.50e21·9-s − 3.55e21·10-s − 1.02e23·11-s + 6.70e23·12-s + 1.50e25·13-s + 1.78e25·14-s + 3.23e25·15-s + 3.09e26·16-s + 1.34e27·17-s + 6.30e27·18-s − 8.78e28·19-s + 1.49e28·20-s − 1.61e29·21-s + 4.28e29·22-s + 1.71e30·23-s − 2.81e30·24-s − 2.77e31·25-s − 6.30e31·26-s − 1.69e32·27-s − 7.47e31·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.700·3-s + 0.5·4-s + 0.159·5-s − 0.495·6-s − 0.410·7-s − 0.353·8-s − 0.508·9-s − 0.112·10-s − 0.378·11-s + 0.350·12-s + 1.29·13-s + 0.290·14-s + 0.111·15-s + 0.250·16-s + 0.276·17-s + 0.359·18-s − 1.48·19-s + 0.0795·20-s − 0.287·21-s + 0.267·22-s + 0.392·23-s − 0.247·24-s − 0.974·25-s − 0.918·26-s − 1.05·27-s − 0.205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(46-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+45/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(23)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{47}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.19e6T \) |
| good | 3 | \( 1 - 3.80e10T + 2.95e21T^{2} \) |
| 5 | \( 1 - 8.48e14T + 2.84e31T^{2} \) |
| 7 | \( 1 + 4.24e18T + 1.07e38T^{2} \) |
| 11 | \( 1 + 1.02e23T + 7.28e46T^{2} \) |
| 13 | \( 1 - 1.50e25T + 1.34e50T^{2} \) |
| 17 | \( 1 - 1.34e27T + 2.34e55T^{2} \) |
| 19 | \( 1 + 8.78e28T + 3.49e57T^{2} \) |
| 23 | \( 1 - 1.71e30T + 1.89e61T^{2} \) |
| 29 | \( 1 + 8.95e32T + 6.42e65T^{2} \) |
| 31 | \( 1 + 4.71e33T + 1.29e67T^{2} \) |
| 37 | \( 1 - 2.99e35T + 3.70e70T^{2} \) |
| 41 | \( 1 + 1.46e36T + 3.76e72T^{2} \) |
| 43 | \( 1 + 1.84e36T + 3.20e73T^{2} \) |
| 47 | \( 1 + 4.82e37T + 1.75e75T^{2} \) |
| 53 | \( 1 - 4.74e38T + 3.91e77T^{2} \) |
| 59 | \( 1 + 9.44e39T + 4.87e79T^{2} \) |
| 61 | \( 1 + 2.42e40T + 2.18e80T^{2} \) |
| 67 | \( 1 - 1.14e40T + 1.49e82T^{2} \) |
| 71 | \( 1 - 5.38e41T + 2.02e83T^{2} \) |
| 73 | \( 1 + 5.81e41T + 7.07e83T^{2} \) |
| 79 | \( 1 - 8.48e42T + 2.47e85T^{2} \) |
| 83 | \( 1 - 1.34e43T + 2.28e86T^{2} \) |
| 89 | \( 1 + 4.34e43T + 5.27e87T^{2} \) |
| 97 | \( 1 - 2.16e44T + 2.53e89T^{2} \) |
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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
−16.70119422864680159108475271424, −15.02535468795523231500746222682, −13.22392564693127267292735812379, −11.00932702424303248942582420852, −9.266449958546842983251100272346, −8.036944833949609739457064015359, −6.08841909311783733854143528588, −3.44388743437458894349875015741, −1.91878378318742479337122401395, 0,
1.91878378318742479337122401395, 3.44388743437458894349875015741, 6.08841909311783733854143528588, 8.036944833949609739457064015359, 9.266449958546842983251100272346, 11.00932702424303248942582420852, 13.22392564693127267292735812379, 15.02535468795523231500746222682, 16.70119422864680159108475271424
