| L(s) = 1 | − 2.93·2-s − 23.3·4-s + 91.0·5-s − 103.·7-s + 162.·8-s − 267.·10-s − 213.·11-s − 479.·13-s + 305.·14-s + 269.·16-s + 838.·17-s − 413.·19-s − 2.12e3·20-s + 626.·22-s − 529·23-s + 5.16e3·25-s + 1.40e3·26-s + 2.42e3·28-s + 853.·29-s + 8.24e3·31-s − 5.99e3·32-s − 2.46e3·34-s − 9.46e3·35-s − 1.59e4·37-s + 1.21e3·38-s + 1.48e4·40-s − 1.33e4·41-s + ⋯ |
| L(s) = 1 | − 0.519·2-s − 0.730·4-s + 1.62·5-s − 0.801·7-s + 0.898·8-s − 0.846·10-s − 0.531·11-s − 0.787·13-s + 0.416·14-s + 0.262·16-s + 0.704·17-s − 0.262·19-s − 1.18·20-s + 0.276·22-s − 0.208·23-s + 1.65·25-s + 0.409·26-s + 0.585·28-s + 0.188·29-s + 1.54·31-s − 1.03·32-s − 0.365·34-s − 1.30·35-s − 1.91·37-s + 0.136·38-s + 1.46·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 2.93T + 32T^{2} \) |
| 5 | \( 1 - 91.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 103.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 213.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 479.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 838.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 413.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 853.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.59e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.37e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.69e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.06e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.55e4T + 8.58e9T^{2} \) |
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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
−10.33113478349351951432547896347, −10.06226262754975518128484997932, −9.279976628485532503516574938109, −8.257346427039392499619665594832, −6.88715095035605820809532264353, −5.74023809650753242525038457219, −4.80675518587349494167004087120, −2.99092606859396436636211723765, −1.57561929098258746583385356560, 0,
1.57561929098258746583385356560, 2.99092606859396436636211723765, 4.80675518587349494167004087120, 5.74023809650753242525038457219, 6.88715095035605820809532264353, 8.257346427039392499619665594832, 9.279976628485532503516574938109, 10.06226262754975518128484997932, 10.33113478349351951432547896347
