| L(s) = 1 | + 3.47·2-s + 11.9·3-s − 19.8·4-s + 25·5-s + 41.7·6-s − 244.·7-s − 180.·8-s − 99.2·9-s + 86.9·10-s − 121·11-s − 238.·12-s − 120.·13-s − 849.·14-s + 299.·15-s + 8.20·16-s − 448.·17-s − 345.·18-s − 361·19-s − 497.·20-s − 2.92e3·21-s − 421.·22-s − 1.68e3·23-s − 2.16e3·24-s + 625·25-s − 419.·26-s − 4.10e3·27-s + 4.85e3·28-s + ⋯ |
| L(s) = 1 | + 0.615·2-s + 0.769·3-s − 0.621·4-s + 0.447·5-s + 0.473·6-s − 1.88·7-s − 0.997·8-s − 0.408·9-s + 0.275·10-s − 0.301·11-s − 0.478·12-s − 0.197·13-s − 1.15·14-s + 0.343·15-s + 0.00801·16-s − 0.376·17-s − 0.251·18-s − 0.229·19-s − 0.277·20-s − 1.44·21-s − 0.185·22-s − 0.663·23-s − 0.767·24-s + 0.200·25-s − 0.121·26-s − 1.08·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.025132261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025132261\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 3.47T + 32T^{2} \) |
| 3 | \( 1 - 11.9T + 243T^{2} \) |
| 7 | \( 1 + 244.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 120.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 448.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.68e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.49e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.94e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.65e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.41e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.03e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.32e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.28e4T + 8.58e9T^{2} \) |
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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
−9.152415018931888773785052364747, −8.667616407276866064699434791003, −7.52614142742475278050440153158, −6.32650125171039098057480737709, −5.91834976907233893761452884709, −4.78625815844028511095079427072, −3.63616377194007006448676252820, −3.12375391850150563242791569696, −2.24889902371588917929694928713, −0.35570929534425394793987069957,
0.35570929534425394793987069957, 2.24889902371588917929694928713, 3.12375391850150563242791569696, 3.63616377194007006448676252820, 4.78625815844028511095079427072, 5.91834976907233893761452884709, 6.32650125171039098057480737709, 7.52614142742475278050440153158, 8.667616407276866064699434791003, 9.152415018931888773785052364747
