| L(s) = 1 | − 3.68·2-s − 20.7·3-s − 18.3·4-s − 25·5-s + 76.5·6-s + 188.·7-s + 185.·8-s + 187.·9-s + 92.2·10-s + 121·11-s + 381.·12-s − 159.·13-s − 694.·14-s + 518.·15-s − 97.1·16-s + 1.50e3·17-s − 691.·18-s + 361·19-s + 459.·20-s − 3.90e3·21-s − 446.·22-s − 4.27e3·23-s − 3.85e3·24-s + 625·25-s + 588.·26-s + 1.15e3·27-s − 3.46e3·28-s + ⋯ |
| L(s) = 1 | − 0.652·2-s − 1.33·3-s − 0.574·4-s − 0.447·5-s + 0.867·6-s + 1.45·7-s + 1.02·8-s + 0.770·9-s + 0.291·10-s + 0.301·11-s + 0.764·12-s − 0.261·13-s − 0.947·14-s + 0.595·15-s − 0.0948·16-s + 1.26·17-s − 0.502·18-s + 0.229·19-s + 0.257·20-s − 1.93·21-s − 0.196·22-s − 1.68·23-s − 1.36·24-s + 0.200·25-s + 0.170·26-s + 0.304·27-s − 0.835·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 3.68T + 32T^{2} \) |
| 3 | \( 1 + 20.7T + 243T^{2} \) |
| 7 | \( 1 - 188.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 159.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.50e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 396.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 361.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.66e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.54e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.74e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.52e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.47e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.69e5T + 8.58e9T^{2} \) |
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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
−8.598401312257007367144552067729, −7.995222234780695183979052654817, −7.35178979183154608081112184096, −6.10822523740137505128185322874, −5.18061513780080802946028169644, −4.70831228726304808121281989190, −3.70641690961173404689815656577, −1.75386115594595928250846394817, −0.932152215502840940113243574937, 0,
0.932152215502840940113243574937, 1.75386115594595928250846394817, 3.70641690961173404689815656577, 4.70831228726304808121281989190, 5.18061513780080802946028169644, 6.10822523740137505128185322874, 7.35178979183154608081112184096, 7.995222234780695183979052654817, 8.598401312257007367144552067729
