| L(s) = 1 | + 1.30·2-s − 3-s + 0.715·4-s − 0.830·5-s − 1.30·6-s + 1.68·7-s − 0.372·8-s + 9-s − 1.08·10-s + 1.91·11-s − 0.715·12-s + 2.20·14-s + 0.830·15-s − 1.20·16-s + 0.284·17-s + 1.30·18-s − 1.91·19-s − 0.594·20-s − 1.68·21-s + 2.51·22-s + 0.372·24-s − 0.309·25-s − 27-s + 1.20·28-s + 1.08·30-s − 1.20·32-s − 1.91·33-s + ⋯ |
| L(s) = 1 | + 1.30·2-s − 3-s + 0.715·4-s − 0.830·5-s − 1.30·6-s + 1.68·7-s − 0.372·8-s + 9-s − 1.08·10-s + 1.91·11-s − 0.715·12-s + 2.20·14-s + 0.830·15-s − 1.20·16-s + 0.284·17-s + 1.30·18-s − 1.91·19-s − 0.594·20-s − 1.68·21-s + 2.51·22-s + 0.372·24-s − 0.309·25-s − 27-s + 1.20·28-s + 1.08·30-s − 1.20·32-s − 1.91·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 591 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 591 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.304260421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304260421\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 - 1.30T + T^{2} \) |
| 5 | \( 1 + 0.830T + T^{2} \) |
| 7 | \( 1 - 1.68T + T^{2} \) |
| 11 | \( 1 - 1.91T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.284T + T^{2} \) |
| 19 | \( 1 + 1.91T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.284T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.30T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.830T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.68T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.30T + T^{2} \) |
| 97 | \( 1 + 0.284T + T^{2} \) |
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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
−11.46660446381358222216219000715, −10.52681210617143193153796370477, −9.043391577754183801151534251808, −8.131165755643987124287911486892, −6.92935312768088336240933509944, −6.19623899487199539124863349461, −5.13051133890941151135768396585, −4.28591092828602881799368338735, −3.92019783737823609810945523924, −1.71661130588285073503029844428,
1.71661130588285073503029844428, 3.92019783737823609810945523924, 4.28591092828602881799368338735, 5.13051133890941151135768396585, 6.19623899487199539124863349461, 6.92935312768088336240933509944, 8.131165755643987124287911486892, 9.043391577754183801151534251808, 10.52681210617143193153796370477, 11.46660446381358222216219000715
