| L(s) = 1 | + 6.94·2-s + 12.2·3-s + 16.2·4-s − 25·5-s + 85.3·6-s + 133.·7-s − 109.·8-s − 91.8·9-s − 173.·10-s − 121·11-s + 199.·12-s + 61.7·13-s + 926.·14-s − 307.·15-s − 1.27e3·16-s + 1.37e3·17-s − 638.·18-s − 361·19-s − 405.·20-s + 1.63e3·21-s − 840.·22-s + 3.67e3·23-s − 1.34e3·24-s + 625·25-s + 428.·26-s − 4.11e3·27-s + 2.16e3·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.788·3-s + 0.507·4-s − 0.447·5-s + 0.968·6-s + 1.02·7-s − 0.605·8-s − 0.378·9-s − 0.549·10-s − 0.301·11-s + 0.399·12-s + 0.101·13-s + 1.26·14-s − 0.352·15-s − 1.24·16-s + 1.15·17-s − 0.464·18-s − 0.229·19-s − 0.226·20-s + 0.811·21-s − 0.370·22-s + 1.45·23-s − 0.477·24-s + 0.200·25-s + 0.124·26-s − 1.08·27-s + 0.521·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 6.94T + 32T^{2} \) |
| 3 | \( 1 - 12.2T + 243T^{2} \) |
| 7 | \( 1 - 133.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 61.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.81e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.37e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 93.8T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.00e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.18e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.95e4T + 8.58e9T^{2} \) |
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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
−8.860788374271424357659880147072, −7.77746310875416788442074617252, −7.28647751054269273278721325447, −5.73698228961109499614876402642, −5.32534598600100489752497997946, −4.27313133832888225194653523421, −3.49084514643723182730342561348, −2.75494994816337707288499549500, −1.59260773712124852090818321328, 0,
1.59260773712124852090818321328, 2.75494994816337707288499549500, 3.49084514643723182730342561348, 4.27313133832888225194653523421, 5.32534598600100489752497997946, 5.73698228961109499614876402642, 7.28647751054269273278721325447, 7.77746310875416788442074617252, 8.860788374271424357659880147072
