| L(s) = 1 | + 2.22·2-s + 1.90·3-s + 2.93·4-s − 5-s + 4.22·6-s + 7-s + 2.07·8-s + 0.612·9-s − 2.22·10-s − 4.55·11-s + 5.57·12-s + 1.25·13-s + 2.22·14-s − 1.90·15-s − 1.26·16-s − 4.51·17-s + 1.36·18-s − 6.84·19-s − 2.93·20-s + 1.90·21-s − 10.1·22-s + 0.652·23-s + 3.94·24-s + 25-s + 2.78·26-s − 4.53·27-s + 2.93·28-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.09·3-s + 1.46·4-s − 0.447·5-s + 1.72·6-s + 0.377·7-s + 0.732·8-s + 0.204·9-s − 0.702·10-s − 1.37·11-s + 1.60·12-s + 0.347·13-s + 0.593·14-s − 0.490·15-s − 0.315·16-s − 1.09·17-s + 0.320·18-s − 1.57·19-s − 0.655·20-s + 0.414·21-s − 2.15·22-s + 0.136·23-s + 0.804·24-s + 0.200·25-s + 0.545·26-s − 0.873·27-s + 0.554·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.22T + 2T^{2} \) |
| 3 | \( 1 - 1.90T + 3T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 - 1.25T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 - 0.652T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 37 | \( 1 + 5.03T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 - 5.01T + 53T^{2} \) |
| 59 | \( 1 + 6.41T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 8.81T + 71T^{2} \) |
| 73 | \( 1 + 5.22T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 1.33T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38438264829910198048739843229, −6.79536170327731459899787416166, −5.81585330927612763206272793718, −5.33988026161625539936442259376, −4.37492778594512456187003610013, −4.02969819631276752750761074555, −3.21433721495088617153136626775, −2.41591790926960654618685610459, −2.06691109004490930951448342430, 0,
2.06691109004490930951448342430, 2.41591790926960654618685610459, 3.21433721495088617153136626775, 4.02969819631276752750761074555, 4.37492778594512456187003610013, 5.33988026161625539936442259376, 5.81585330927612763206272793718, 6.79536170327731459899787416166, 7.38438264829910198048739843229
