| L(s) = 1 | + 2.42·2-s + 1.02·3-s + 3.89·4-s − 5-s + 2.49·6-s − 7-s + 4.59·8-s − 1.94·9-s − 2.42·10-s + 0.740·11-s + 3.99·12-s − 0.862·13-s − 2.42·14-s − 1.02·15-s + 3.36·16-s + 2.25·17-s − 4.72·18-s + 4.76·19-s − 3.89·20-s − 1.02·21-s + 1.79·22-s − 0.0894·23-s + 4.71·24-s + 25-s − 2.09·26-s − 5.07·27-s − 3.89·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.592·3-s + 1.94·4-s − 0.447·5-s + 1.01·6-s − 0.377·7-s + 1.62·8-s − 0.648·9-s − 0.767·10-s + 0.223·11-s + 1.15·12-s − 0.239·13-s − 0.648·14-s − 0.265·15-s + 0.842·16-s + 0.545·17-s − 1.11·18-s + 1.09·19-s − 0.870·20-s − 0.224·21-s + 0.383·22-s − 0.0186·23-s + 0.963·24-s + 0.200·25-s − 0.410·26-s − 0.977·27-s − 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.686006312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.686006312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 - 1.02T + 3T^{2} \) |
| 11 | \( 1 - 0.740T + 11T^{2} \) |
| 13 | \( 1 + 0.862T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 + 0.0894T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 7.27T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 - 1.00T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 0.888T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 - 6.37T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 9.82T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 8.20T + 79T^{2} \) |
| 83 | \( 1 - 5.30T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.56232091226411235190672140431, −7.05993299709732078855311310440, −6.18759238502686346701633386743, −5.71416404239413784104637660898, −4.89893099531415662115592188851, −4.25115354954441943449311050672, −3.49911790365300277361095770023, −2.89520619893671893622910959588, −2.44932797289506742975946412520, −0.966307561496489971119275629044,
0.966307561496489971119275629044, 2.44932797289506742975946412520, 2.89520619893671893622910959588, 3.49911790365300277361095770023, 4.25115354954441943449311050672, 4.89893099531415662115592188851, 5.71416404239413784104637660898, 6.18759238502686346701633386743, 7.05993299709732078855311310440, 7.56232091226411235190672140431
