| L(s) = 1 | − 2-s + 2.27·3-s + 4-s − 1.29·5-s − 2.27·6-s − 2.53·7-s − 8-s + 2.16·9-s + 1.29·10-s − 0.130·11-s + 2.27·12-s − 0.185·13-s + 2.53·14-s − 2.94·15-s + 16-s − 0.0597·17-s − 2.16·18-s − 4.43·19-s − 1.29·20-s − 5.76·21-s + 0.130·22-s − 2.35·23-s − 2.27·24-s − 3.32·25-s + 0.185·26-s − 1.89·27-s − 2.53·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.31·3-s + 0.5·4-s − 0.579·5-s − 0.928·6-s − 0.958·7-s − 0.353·8-s + 0.722·9-s + 0.409·10-s − 0.0393·11-s + 0.656·12-s − 0.0515·13-s + 0.677·14-s − 0.760·15-s + 0.250·16-s − 0.0144·17-s − 0.511·18-s − 1.01·19-s − 0.289·20-s − 1.25·21-s + 0.0278·22-s − 0.491·23-s − 0.464·24-s − 0.664·25-s + 0.0364·26-s − 0.363·27-s − 0.479·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 + 0.130T + 11T^{2} \) |
| 13 | \( 1 + 0.185T + 13T^{2} \) |
| 17 | \( 1 + 0.0597T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 31 | \( 1 - 8.33T + 31T^{2} \) |
| 37 | \( 1 + 9.75T + 37T^{2} \) |
| 41 | \( 1 + 6.74T + 41T^{2} \) |
| 43 | \( 1 + 0.618T + 43T^{2} \) |
| 47 | \( 1 + 9.33T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 1.15T + 83T^{2} \) |
| 89 | \( 1 - 6.87T + 89T^{2} \) |
| 97 | \( 1 + 6.52T + 97T^{2} \) |
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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
−8.868309585184943275268424799899, −8.376913292273316499008200256055, −7.82121604671594229966327154156, −6.81337694574586400886062018580, −6.19819174438780100713157685972, −4.63618151711952042568864084760, −3.53875215935714061926334580474, −2.94242333830367144346149977731, −1.85931233632875258013399521483, 0,
1.85931233632875258013399521483, 2.94242333830367144346149977731, 3.53875215935714061926334580474, 4.63618151711952042568864084760, 6.19819174438780100713157685972, 6.81337694574586400886062018580, 7.82121604671594229966327154156, 8.376913292273316499008200256055, 8.868309585184943275268424799899
