| L(s) = 1 | − 5.96·2-s + 9·3-s + 3.52·4-s + 65.6·5-s − 53.6·6-s + 119.·7-s + 169.·8-s + 81·9-s − 391.·10-s + 237.·11-s + 31.7·12-s + 999.·13-s − 709.·14-s + 590.·15-s − 1.12e3·16-s − 1.27e3·17-s − 482.·18-s + 1.42e3·19-s + 231.·20-s + 1.07e3·21-s − 1.41e3·22-s + 546.·23-s + 1.52e3·24-s + 1.18e3·25-s − 5.95e3·26-s + 729·27-s + 419.·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.577·3-s + 0.110·4-s + 1.17·5-s − 0.608·6-s + 0.918·7-s + 0.937·8-s + 0.333·9-s − 1.23·10-s + 0.592·11-s + 0.0635·12-s + 1.63·13-s − 0.967·14-s + 0.677·15-s − 1.09·16-s − 1.07·17-s − 0.351·18-s + 0.906·19-s + 0.129·20-s + 0.530·21-s − 0.623·22-s + 0.215·23-s + 0.541·24-s + 0.378·25-s − 1.72·26-s + 0.192·27-s + 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.023806391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023806391\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 + 5.96T + 32T^{2} \) |
| 5 | \( 1 - 65.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 119.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 237.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 999.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.27e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 546.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.90e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 7.23e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.08e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.17e5T + 8.58e9T^{2} \) |
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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
−11.34194255764223341647508751966, −10.62476041256158269869529120292, −9.430196724757437899891888399993, −8.944130306332400848352492250360, −8.062018692526054879521726836954, −6.81715657767036138761339413074, −5.42727246619447774829103829280, −3.96616703495734610580077691878, −1.97859293909277273559541661128, −1.17858681667078262042467960945,
1.17858681667078262042467960945, 1.97859293909277273559541661128, 3.96616703495734610580077691878, 5.42727246619447774829103829280, 6.81715657767036138761339413074, 8.062018692526054879521726836954, 8.944130306332400848352492250360, 9.430196724757437899891888399993, 10.62476041256158269869529120292, 11.34194255764223341647508751966
