| L(s) = 1 | − 1.08·2-s − 0.817·4-s − 0.209·5-s + 7-s + 3.06·8-s + 0.227·10-s − 6.03·11-s + 3.67·13-s − 1.08·14-s − 1.69·16-s + 5.37·17-s − 3.54·19-s + 0.171·20-s + 6.56·22-s + 1.30·23-s − 4.95·25-s − 3.99·26-s − 0.817·28-s + 8.00·29-s − 0.384·31-s − 4.28·32-s − 5.84·34-s − 0.209·35-s − 3.68·37-s + 3.85·38-s − 0.642·40-s + 41-s + ⋯ |
| L(s) = 1 | − 0.768·2-s − 0.408·4-s − 0.0937·5-s + 0.377·7-s + 1.08·8-s + 0.0720·10-s − 1.82·11-s + 1.01·13-s − 0.290·14-s − 0.423·16-s + 1.30·17-s − 0.812·19-s + 0.0383·20-s + 1.39·22-s + 0.271·23-s − 0.991·25-s − 0.782·26-s − 0.154·28-s + 1.48·29-s − 0.0690·31-s − 0.757·32-s − 1.00·34-s − 0.0354·35-s − 0.606·37-s + 0.624·38-s − 0.101·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8784052149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8784052149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.08T + 2T^{2} \) |
| 5 | \( 1 + 0.209T + 5T^{2} \) |
| 11 | \( 1 + 6.03T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 - 8.00T + 29T^{2} \) |
| 31 | \( 1 + 0.384T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 43 | \( 1 - 0.824T + 43T^{2} \) |
| 47 | \( 1 + 5.11T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 + 6.04T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 0.520T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.682202539801133476258777132984, −8.061843340696142761475003550040, −7.87700319000645031972736428618, −6.76170789771864740247009202275, −5.63592088304632199184764650069, −5.05356947271252682217065968634, −4.12998784895260265748326295629, −3.10154714541824482289137422632, −1.87836603040906460612751222704, −0.67432600640433240134757344861,
0.67432600640433240134757344861, 1.87836603040906460612751222704, 3.10154714541824482289137422632, 4.12998784895260265748326295629, 5.05356947271252682217065968634, 5.63592088304632199184764650069, 6.76170789771864740247009202275, 7.87700319000645031972736428618, 8.061843340696142761475003550040, 8.682202539801133476258777132984
