| L(s) = 1 | − 2-s − 1.97·3-s + 4-s + 2.94·5-s + 1.97·6-s − 0.656·7-s − 8-s + 0.915·9-s − 2.94·10-s + 11-s − 1.97·12-s − 0.457·13-s + 0.656·14-s − 5.83·15-s + 16-s + 1.53·17-s − 0.915·18-s + 2.94·20-s + 1.30·21-s − 22-s + 3.48·23-s + 1.97·24-s + 3.68·25-s + 0.457·26-s + 4.12·27-s − 0.656·28-s − 4.19·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.14·3-s + 0.5·4-s + 1.31·5-s + 0.807·6-s − 0.248·7-s − 0.353·8-s + 0.305·9-s − 0.931·10-s + 0.301·11-s − 0.571·12-s − 0.126·13-s + 0.175·14-s − 1.50·15-s + 0.250·16-s + 0.372·17-s − 0.215·18-s + 0.658·20-s + 0.283·21-s − 0.213·22-s + 0.726·23-s + 0.403·24-s + 0.736·25-s + 0.0896·26-s + 0.793·27-s − 0.124·28-s − 0.778·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.97T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 + 0.656T + 7T^{2} \) |
| 13 | \( 1 + 0.457T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 + 8.04T + 31T^{2} \) |
| 37 | \( 1 + 0.651T + 37T^{2} \) |
| 41 | \( 1 + 5.02T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 3.32T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 - 5.52T + 61T^{2} \) |
| 67 | \( 1 - 7.85T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 - 1.39T + 89T^{2} \) |
| 97 | \( 1 - 1.04T + 97T^{2} \) |
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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
−7.24284408800072043811516968623, −6.75822656837895857089195406521, −6.13534289838591397213196295049, −5.42268390600607130181743986431, −5.21222741093111166066064759142, −3.88943093819181420207570564336, −2.88383461946020953760678378269, −1.94298194622249257017862126524, −1.16075475652675153820530839515, 0,
1.16075475652675153820530839515, 1.94298194622249257017862126524, 2.88383461946020953760678378269, 3.88943093819181420207570564336, 5.21222741093111166066064759142, 5.42268390600607130181743986431, 6.13534289838591397213196295049, 6.75822656837895857089195406521, 7.24284408800072043811516968623
