| L(s) = 1 | + 3.67·2-s + 5.49·4-s + 5·5-s − 13.6·7-s − 9.19·8-s + 18.3·10-s − 11·11-s − 32.0·13-s − 49.9·14-s − 77.7·16-s − 100.·17-s + 5.21·19-s + 27.4·20-s − 40.4·22-s − 62.2·23-s + 25·25-s − 117.·26-s − 74.7·28-s + 124.·29-s + 98.9·31-s − 212.·32-s − 369.·34-s − 68.0·35-s + 32.0·37-s + 19.1·38-s − 45.9·40-s − 136.·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.687·4-s + 0.447·5-s − 0.734·7-s − 0.406·8-s + 0.580·10-s − 0.301·11-s − 0.683·13-s − 0.954·14-s − 1.21·16-s − 1.43·17-s + 0.0629·19-s + 0.307·20-s − 0.391·22-s − 0.564·23-s + 0.200·25-s − 0.887·26-s − 0.504·28-s + 0.799·29-s + 0.573·31-s − 1.17·32-s − 1.86·34-s − 0.328·35-s + 0.142·37-s + 0.0817·38-s − 0.181·40-s − 0.518·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.67T + 8T^{2} \) |
| 7 | \( 1 + 13.6T + 343T^{2} \) |
| 13 | \( 1 + 32.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.21T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 98.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 32.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 159.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 382.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 201.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 196.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 202.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 670.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 376.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 745.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.74e3T + 9.12e5T^{2} \) |
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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
−10.08954965447306475048425136380, −9.350915082379679272583768583898, −8.286614575669794239237804710939, −6.80313962586339735609578828309, −6.29725765505595323081358581341, −5.17958037877326600475525141528, −4.40397050461932551493210030692, −3.19336113261014207695790509742, −2.24936982179419902427804131714, 0,
2.24936982179419902427804131714, 3.19336113261014207695790509742, 4.40397050461932551493210030692, 5.17958037877326600475525141528, 6.29725765505595323081358581341, 6.80313962586339735609578828309, 8.286614575669794239237804710939, 9.350915082379679272583768583898, 10.08954965447306475048425136380
