| L(s) = 1 | − 2.33·2-s − 2.30·3-s + 3.43·4-s − 3.37·5-s + 5.38·6-s − 3.34·8-s + 2.33·9-s + 7.85·10-s + 2.33·11-s − 7.93·12-s + 7.78·15-s + 0.933·16-s + 5.45·17-s − 5.43·18-s + 7.16·19-s − 11.5·20-s − 5.43·22-s − 6.45·23-s + 7.73·24-s + 6.36·25-s + 1.54·27-s − 8.44·29-s − 18.1·30-s + 3.05·31-s + 4.51·32-s − 5.38·33-s − 12.7·34-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 1.33·3-s + 1.71·4-s − 1.50·5-s + 2.19·6-s − 1.18·8-s + 0.777·9-s + 2.48·10-s + 0.702·11-s − 2.29·12-s + 2.00·15-s + 0.233·16-s + 1.32·17-s − 1.28·18-s + 1.64·19-s − 2.59·20-s − 1.15·22-s − 1.34·23-s + 1.57·24-s + 1.27·25-s + 0.297·27-s − 1.56·29-s − 3.31·30-s + 0.548·31-s + 0.798·32-s − 0.937·33-s − 2.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 - 7.16T + 19T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 - 0.937T + 41T^{2} \) |
| 43 | \( 1 + 4.09T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 + 7.24T + 59T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 + 7.58T + 71T^{2} \) |
| 73 | \( 1 - 2.06T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 - 2.89T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 3.55T + 97T^{2} \) |
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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
−7.56061587951193583221063963250, −7.14338302481149401918303201288, −6.21955420735550428486633560033, −5.64959263499632331214619971486, −4.70111761974110096811226866759, −3.85195971539605304429996585604, −3.06056664899500846314100733943, −1.54893250462767763179822117152, −0.816178458964244307761123419204, 0,
0.816178458964244307761123419204, 1.54893250462767763179822117152, 3.06056664899500846314100733943, 3.85195971539605304429996585604, 4.70111761974110096811226866759, 5.64959263499632331214619971486, 6.21955420735550428486633560033, 7.14338302481149401918303201288, 7.56061587951193583221063963250
