| L(s) = 1 | + 0.634·2-s + 2.14·3-s − 1.59·4-s + 5-s + 1.36·6-s − 2.28·8-s + 1.59·9-s + 0.634·10-s − 4.04·11-s − 3.42·12-s + 2.94·13-s + 2.14·15-s + 1.74·16-s + 0.682·17-s + 1.01·18-s − 1.02·19-s − 1.59·20-s − 2.56·22-s + 1.64·23-s − 4.89·24-s + 25-s + 1.86·26-s − 3.01·27-s − 3.28·29-s + 1.36·30-s − 31-s + 5.67·32-s + ⋯ |
| L(s) = 1 | + 0.448·2-s + 1.23·3-s − 0.798·4-s + 0.447·5-s + 0.555·6-s − 0.807·8-s + 0.531·9-s + 0.200·10-s − 1.22·11-s − 0.988·12-s + 0.816·13-s + 0.553·15-s + 0.436·16-s + 0.165·17-s + 0.238·18-s − 0.234·19-s − 0.357·20-s − 0.547·22-s + 0.343·23-s − 0.998·24-s + 0.200·25-s + 0.366·26-s − 0.580·27-s − 0.609·29-s + 0.248·30-s − 0.179·31-s + 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.634T + 2T^{2} \) |
| 3 | \( 1 - 2.14T + 3T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 - 0.682T + 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 + 9.07T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 - 0.358T + 59T^{2} \) |
| 61 | \( 1 + 4.59T + 61T^{2} \) |
| 67 | \( 1 - 4.46T + 67T^{2} \) |
| 71 | \( 1 + 4.92T + 71T^{2} \) |
| 73 | \( 1 + 5.29T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 - 4.00T + 83T^{2} \) |
| 89 | \( 1 + 5.32T + 89T^{2} \) |
| 97 | \( 1 - 5.83T + 97T^{2} \) |
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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
−7.82418494796290877731172160764, −6.85825018506115827411136594296, −5.96936760929453892181706308342, −5.31515826554876198865271348665, −4.70763689572456599364625886997, −3.68944666578223644680953438569, −3.25572854886252976824086326154, −2.48846414315526098567656886987, −1.52523124960233970134818709266, 0,
1.52523124960233970134818709266, 2.48846414315526098567656886987, 3.25572854886252976824086326154, 3.68944666578223644680953438569, 4.70763689572456599364625886997, 5.31515826554876198865271348665, 5.96936760929453892181706308342, 6.85825018506115827411136594296, 7.82418494796290877731172160764
