| L(s) = 1 | − 8.11·2-s − 3.22·3-s + 33.8·4-s + 26.2·6-s − 20.7·7-s − 15.2·8-s − 232.·9-s + 134.·11-s − 109.·12-s + 169·13-s + 168.·14-s − 960.·16-s + 2.19e3·17-s + 1.88e3·18-s − 1.98e3·19-s + 66.9·21-s − 1.09e3·22-s − 4.22e3·23-s + 49.3·24-s − 1.37e3·26-s + 1.53e3·27-s − 702.·28-s + 2.54e3·29-s − 1.36e3·31-s + 8.28e3·32-s − 435.·33-s − 1.77e4·34-s + ⋯ |
| L(s) = 1 | − 1.43·2-s − 0.207·3-s + 1.05·4-s + 0.297·6-s − 0.159·7-s − 0.0844·8-s − 0.957·9-s + 0.335·11-s − 0.219·12-s + 0.277·13-s + 0.229·14-s − 0.937·16-s + 1.84·17-s + 1.37·18-s − 1.25·19-s + 0.0331·21-s − 0.481·22-s − 1.66·23-s + 0.0175·24-s − 0.397·26-s + 0.405·27-s − 0.169·28-s + 0.562·29-s − 0.255·31-s + 1.42·32-s − 0.0695·33-s − 2.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 8.11T + 32T^{2} \) |
| 3 | \( 1 + 3.22T + 243T^{2} \) |
| 7 | \( 1 + 20.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 134.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 2.19e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.98e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.41e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.43e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.21e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.13e4T + 8.58e9T^{2} \) |
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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
−10.10450339333583716460034770096, −9.443630240863781078387908780103, −8.318377963252344014759387160590, −7.915944262158641017008681802680, −6.55021406796246856390223747174, −5.67807571886183690224958675700, −4.05808330657680428863465895109, −2.51764199122201399801349149876, −1.13114352382471593720895738788, 0,
1.13114352382471593720895738788, 2.51764199122201399801349149876, 4.05808330657680428863465895109, 5.67807571886183690224958675700, 6.55021406796246856390223747174, 7.915944262158641017008681802680, 8.318377963252344014759387160590, 9.443630240863781078387908780103, 10.10450339333583716460034770096
