| L(s) = 1 | + 2.07·2-s − 1.05·3-s + 2.32·4-s − 2.22·5-s − 2.19·6-s + 0.673·8-s − 1.88·9-s − 4.61·10-s + 0.863·11-s − 2.45·12-s − 3.79·13-s + 2.34·15-s − 3.24·16-s − 17-s − 3.91·18-s − 2.16·19-s − 5.16·20-s + 1.79·22-s − 3.24·23-s − 0.710·24-s − 0.0652·25-s − 7.88·26-s + 5.15·27-s + 4.81·29-s + 4.87·30-s + 1.44·31-s − 8.09·32-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.609·3-s + 1.16·4-s − 0.993·5-s − 0.896·6-s + 0.237·8-s − 0.628·9-s − 1.46·10-s + 0.260·11-s − 0.708·12-s − 1.05·13-s + 0.605·15-s − 0.811·16-s − 0.242·17-s − 0.923·18-s − 0.497·19-s − 1.15·20-s + 0.382·22-s − 0.676·23-s − 0.145·24-s − 0.0130·25-s − 1.54·26-s + 0.992·27-s + 0.894·29-s + 0.890·30-s + 0.259·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 11 | \( 1 - 0.863T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 4.81T + 29T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 + 7.71T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 0.792T + 61T^{2} \) |
| 67 | \( 1 - 1.42T + 67T^{2} \) |
| 71 | \( 1 + 4.93T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 0.0684T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 - 8.33T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−10.04780974681883166555292841939, −8.827571592557884168894622094949, −7.88784203120822417505163531774, −6.83089125920867564070620725347, −6.11921458728824632114582544122, −5.13153716022182583508693821607, −4.46505301789950275286310374157, −3.55091453534463481733155393281, −2.46867623093761424714909481904, 0,
2.46867623093761424714909481904, 3.55091453534463481733155393281, 4.46505301789950275286310374157, 5.13153716022182583508693821607, 6.11921458728824632114582544122, 6.83089125920867564070620725347, 7.88784203120822417505163531774, 8.827571592557884168894622094949, 10.04780974681883166555292841939
