| L(s) = 1 | − 4.47·2-s − 9·3-s − 11.9·4-s + 25·5-s + 40.3·6-s − 168.·7-s + 196.·8-s + 81·9-s − 111.·10-s + 121·11-s + 107.·12-s + 290.·13-s + 756.·14-s − 225·15-s − 499.·16-s + 623.·17-s − 362.·18-s − 398.·19-s − 298.·20-s + 1.51e3·21-s − 541.·22-s + 3.78e3·23-s − 1.77e3·24-s + 625·25-s − 1.30e3·26-s − 729·27-s + 2.01e3·28-s + ⋯ |
| L(s) = 1 | − 0.791·2-s − 0.577·3-s − 0.373·4-s + 0.447·5-s + 0.457·6-s − 1.30·7-s + 1.08·8-s + 0.333·9-s − 0.354·10-s + 0.301·11-s + 0.215·12-s + 0.476·13-s + 1.03·14-s − 0.258·15-s − 0.487·16-s + 0.523·17-s − 0.263·18-s − 0.253·19-s − 0.166·20-s + 0.751·21-s − 0.238·22-s + 1.49·23-s − 0.627·24-s + 0.200·25-s − 0.377·26-s − 0.192·27-s + 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 4.47T + 32T^{2} \) |
| 7 | \( 1 + 168.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 290.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 623.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 398.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 301.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 151.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 476.T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.80e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.16e4T + 8.58e9T^{2} \) |
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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
−11.13341914292434618718985364345, −10.18622369971246515459419562955, −9.490724777722922210731898760876, −8.612606613157344566155336451451, −7.14726641791697984368910402828, −6.19802634034931418836389869135, −4.90553161478220412891672427015, −3.36720458660070725424361496679, −1.29324763946524851849160716602, 0,
1.29324763946524851849160716602, 3.36720458660070725424361496679, 4.90553161478220412891672427015, 6.19802634034931418836389869135, 7.14726641791697984368910402828, 8.612606613157344566155336451451, 9.490724777722922210731898760876, 10.18622369971246515459419562955, 11.13341914292434618718985364345
