| L(s) = 1 | + 3.46·2-s + 3.98·4-s + 6.55·5-s − 13.9·8-s + 22.6·10-s − 2.29·11-s + 28.6·13-s − 79.9·16-s − 46.6·17-s − 67.5·19-s + 26.0·20-s − 7.94·22-s + 30.1·23-s − 82.0·25-s + 99.1·26-s − 24.0·29-s − 193.·31-s − 165.·32-s − 161.·34-s + 208.·37-s − 233.·38-s − 91.2·40-s − 234.·41-s + 46.0·43-s − 9.13·44-s + 104.·46-s + 194.·47-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.497·4-s + 0.586·5-s − 0.614·8-s + 0.717·10-s − 0.0628·11-s + 0.610·13-s − 1.24·16-s − 0.665·17-s − 0.815·19-s + 0.291·20-s − 0.0769·22-s + 0.273·23-s − 0.656·25-s + 0.747·26-s − 0.154·29-s − 1.12·31-s − 0.914·32-s − 0.814·34-s + 0.927·37-s − 0.997·38-s − 0.360·40-s − 0.892·41-s + 0.163·43-s − 0.0312·44-s + 0.334·46-s + 0.602·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.46T + 8T^{2} \) |
| 5 | \( 1 - 6.55T + 125T^{2} \) |
| 11 | \( 1 + 2.29T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 208.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 46.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 221.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 710.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 634.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 269.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 234.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 213.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 242.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
| 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
−8.992728572660845399696065594449, −8.023600173629481591987950923210, −6.82660609143176174487937167169, −6.12953599251508530136288119596, −5.49071525308875413313560328219, −4.53657823815292542868469635494, −3.79994966979357697062035809427, −2.75855447211683108367669926884, −1.73685439566175562739123496783, 0,
1.73685439566175562739123496783, 2.75855447211683108367669926884, 3.79994966979357697062035809427, 4.53657823815292542868469635494, 5.49071525308875413313560328219, 6.12953599251508530136288119596, 6.82660609143176174487937167169, 8.023600173629481591987950923210, 8.992728572660845399696065594449
