| L(s) = 1 | − 6.41·2-s + 9.13·4-s + 25·5-s + 14.3·7-s + 146.·8-s − 160.·10-s − 121·11-s − 434.·13-s − 91.9·14-s − 1.23e3·16-s + 92.0·17-s + 1.96e3·19-s + 228.·20-s + 776.·22-s + 1.12e3·23-s + 625·25-s + 2.78e3·26-s + 130.·28-s − 1.31e3·29-s + 8.13e3·31-s + 3.21e3·32-s − 590.·34-s + 358.·35-s − 5.37e3·37-s − 1.26e4·38-s + 3.66e3·40-s − 1.90e4·41-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.285·4-s + 0.447·5-s + 0.110·7-s + 0.810·8-s − 0.507·10-s − 0.301·11-s − 0.713·13-s − 0.125·14-s − 1.20·16-s + 0.0772·17-s + 1.25·19-s + 0.127·20-s + 0.341·22-s + 0.442·23-s + 0.200·25-s + 0.808·26-s + 0.0315·28-s − 0.289·29-s + 1.52·31-s + 0.554·32-s − 0.0876·34-s + 0.0494·35-s − 0.645·37-s − 1.41·38-s + 0.362·40-s − 1.76·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.052300040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.052300040\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 6.41T + 32T^{2} \) |
| 7 | \( 1 - 14.3T + 1.68e4T^{2} \) |
| 13 | \( 1 + 434.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 92.0T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.96e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.90e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.86e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.03e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.24e5T + 8.58e9T^{2} \) |
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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
−9.934295917879838115928184347263, −9.389713168831522974716883547072, −8.443646378642415310671546482749, −7.62268968099842033618472021353, −6.81628995918393580585490472642, −5.44650022743814476749619361678, −4.57945174191923770224858941070, −2.98209206585992606533066463687, −1.71893689808618620534998478617, −0.63528406611627075302921307779,
0.63528406611627075302921307779, 1.71893689808618620534998478617, 2.98209206585992606533066463687, 4.57945174191923770224858941070, 5.44650022743814476749619361678, 6.81628995918393580585490472642, 7.62268968099842033618472021353, 8.443646378642415310671546482749, 9.389713168831522974716883547072, 9.934295917879838115928184347263
