| L(s) = 1 | + 19.7·2-s + 261.·4-s + 90.5·5-s − 807.·7-s + 2.63e3·8-s + 1.78e3·10-s − 4.97e3·11-s − 2.92e3·13-s − 1.59e4·14-s + 1.85e4·16-s + 2.93e4·17-s + 8.90e3·19-s + 2.36e4·20-s − 9.81e4·22-s + 2.13e4·23-s − 6.99e4·25-s − 5.76e4·26-s − 2.11e5·28-s − 1.44e5·29-s + 1.24e5·31-s + 2.85e4·32-s + 5.78e5·34-s − 7.31e4·35-s + 2.49e5·37-s + 1.75e5·38-s + 2.38e5·40-s − 7.69e5·41-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 2.04·4-s + 0.324·5-s − 0.890·7-s + 1.81·8-s + 0.565·10-s − 1.12·11-s − 0.368·13-s − 1.55·14-s + 1.13·16-s + 1.44·17-s + 0.297·19-s + 0.662·20-s − 1.96·22-s + 0.365·23-s − 0.894·25-s − 0.643·26-s − 1.81·28-s − 1.09·29-s + 0.748·31-s + 0.153·32-s + 2.52·34-s − 0.288·35-s + 0.808·37-s + 0.519·38-s + 0.589·40-s − 1.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 19.7T + 128T^{2} \) |
| 5 | \( 1 - 90.5T + 7.81e4T^{2} \) |
| 7 | \( 1 + 807.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.93e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.90e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.13e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.69e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.14e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.91e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.23e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.01e4T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.76e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.82e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.38e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.52e7T + 8.07e13T^{2} \) |
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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
−9.629107919659658007317462709743, −7.997129825732138904540545291284, −7.14790285581988571004556040628, −6.12711414394672565742474833124, −5.49722488093243794022988339312, −4.68934450650172207956167953093, −3.38791421299365066884623261932, −2.93192720191482530041647200211, −1.72983141344295799680681636428, 0,
1.72983141344295799680681636428, 2.93192720191482530041647200211, 3.38791421299365066884623261932, 4.68934450650172207956167953093, 5.49722488093243794022988339312, 6.12711414394672565742474833124, 7.14790285581988571004556040628, 7.997129825732138904540545291284, 9.629107919659658007317462709743
