| L(s) = 1 | + 4.17·2-s − 2.46·3-s + 9.39·4-s + 7.54·5-s − 10.2·6-s + 5.83·8-s − 20.9·9-s + 31.4·10-s + 26.9·11-s − 23.1·12-s + 15.6·13-s − 18.6·15-s − 50.8·16-s − 27.2·17-s − 87.2·18-s − 38.3·19-s + 70.8·20-s + 112.·22-s + 82.5·23-s − 14.3·24-s − 68.0·25-s + 65.2·26-s + 118.·27-s − 34.2·29-s − 77.6·30-s − 119.·31-s − 258.·32-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.474·3-s + 1.17·4-s + 0.674·5-s − 0.700·6-s + 0.257·8-s − 0.774·9-s + 0.994·10-s + 0.737·11-s − 0.557·12-s + 0.333·13-s − 0.320·15-s − 0.794·16-s − 0.388·17-s − 1.14·18-s − 0.462·19-s + 0.792·20-s + 1.08·22-s + 0.748·23-s − 0.122·24-s − 0.544·25-s + 0.492·26-s + 0.842·27-s − 0.219·29-s − 0.472·30-s − 0.689·31-s − 1.42·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 - 4.17T + 8T^{2} \) |
| 3 | \( 1 + 2.46T + 27T^{2} \) |
| 5 | \( 1 - 7.54T + 125T^{2} \) |
| 11 | \( 1 - 26.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 38.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 34.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 378.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 385.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 906.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 621.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 737.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 558.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.63e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 406.T + 9.12e5T^{2} \) |
| 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
−8.441017583758047904897775543908, −7.13951460350328397271884854460, −6.29093427350591066698088803662, −5.97772172914270141571603880176, −5.16243965145263169117304864571, −4.42226840643435072393778447782, −3.49355427217738791691096423254, −2.63775764402894600203787722337, −1.57191331212374278375954778261, 0,
1.57191331212374278375954778261, 2.63775764402894600203787722337, 3.49355427217738791691096423254, 4.42226840643435072393778447782, 5.16243965145263169117304864571, 5.97772172914270141571603880176, 6.29093427350591066698088803662, 7.13951460350328397271884854460, 8.441017583758047904897775543908
