| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 77.5·5-s + 36·6-s + 95.6·7-s − 64·8-s + 81·9-s + 310.·10-s − 249.·11-s − 144·12-s + 59.0·13-s − 382.·14-s + 698.·15-s + 256·16-s − 1.92e3·17-s − 324·18-s + 1.54e3·19-s − 1.24e3·20-s − 860.·21-s + 998.·22-s + 3.20e3·23-s + 576·24-s + 2.89e3·25-s − 236.·26-s − 729·27-s + 1.52e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.38·5-s + 0.408·6-s + 0.737·7-s − 0.353·8-s + 0.333·9-s + 0.981·10-s − 0.622·11-s − 0.288·12-s + 0.0969·13-s − 0.521·14-s + 0.801·15-s + 0.250·16-s − 1.61·17-s − 0.235·18-s + 0.979·19-s − 0.693·20-s − 0.425·21-s + 0.439·22-s + 1.26·23-s + 0.204·24-s + 0.924·25-s − 0.0685·26-s − 0.192·27-s + 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 + 77.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 95.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 249.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 59.0T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.42e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.69e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 812.T + 4.18e8T^{2} \) |
| 61 | \( 1 + 9.68e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.12e4T + 8.58e9T^{2} \) |
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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
−10.42544875296713009977208518675, −9.107088838907983056360094984732, −8.206190389732280742649267210910, −7.49053284495098937403943639954, −6.62130752787600206094359531439, −5.13438079696508575688335117510, −4.24508336553895002175081796819, −2.75228791162867434869843642854, −1.08212468495584953110107374779, 0,
1.08212468495584953110107374779, 2.75228791162867434869843642854, 4.24508336553895002175081796819, 5.13438079696508575688335117510, 6.62130752787600206094359531439, 7.49053284495098937403943639954, 8.206190389732280742649267210910, 9.107088838907983056360094984732, 10.42544875296713009977208518675
