| L(s) = 1 | − 5.27·2-s − 17.5·3-s − 4.21·4-s − 25·5-s + 92.3·6-s − 132.·7-s + 190.·8-s + 64.1·9-s + 131.·10-s + 73.9·12-s + 62.7·13-s + 696.·14-s + 438.·15-s − 871.·16-s + 112.·17-s − 337.·18-s − 1.78e3·19-s + 105.·20-s + 2.31e3·21-s + 2.24e3·23-s − 3.34e3·24-s + 625·25-s − 330.·26-s + 3.13e3·27-s + 557.·28-s − 14.2·29-s − 2.30e3·30-s + ⋯ |
| L(s) = 1 | − 0.931·2-s − 1.12·3-s − 0.131·4-s − 0.447·5-s + 1.04·6-s − 1.01·7-s + 1.05·8-s + 0.263·9-s + 0.416·10-s + 0.148·12-s + 0.103·13-s + 0.949·14-s + 0.502·15-s − 0.850·16-s + 0.0944·17-s − 0.245·18-s − 1.13·19-s + 0.0589·20-s + 1.14·21-s + 0.884·23-s − 1.18·24-s + 0.200·25-s − 0.0960·26-s + 0.827·27-s + 0.134·28-s − 0.00314·29-s − 0.468·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.007817404712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007817404712\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 5.27T + 32T^{2} \) |
| 3 | \( 1 + 17.5T + 243T^{2} \) |
| 7 | \( 1 + 132.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 62.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 112.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.78e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.24e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 14.2T + 2.05e7T^{2} \) |
| 31 | \( 1 - 307.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.55e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.84e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 774.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.74e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.60e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.73e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−10.02491122562247432010557689016, −8.962855206643238172952233935491, −8.351729837799778193603340947992, −7.10044713725021536843768741109, −6.49204183022593054326520598988, −5.35177933113787458620047980551, −4.42072518152921247606902909652, −3.18608982169978985643879043017, −1.42501401065409282533051825817, −0.05554540686603173249587655211,
0.05554540686603173249587655211, 1.42501401065409282533051825817, 3.18608982169978985643879043017, 4.42072518152921247606902909652, 5.35177933113787458620047980551, 6.49204183022593054326520598988, 7.10044713725021536843768741109, 8.351729837799778193603340947992, 8.962855206643238172952233935491, 10.02491122562247432010557689016
