| L(s) = 1 | − 4·2-s + 28.6·3-s + 16·4-s − 25·5-s − 114.·6-s + 61.5·7-s − 64·8-s + 576.·9-s + 100·10-s − 49.2·11-s + 458.·12-s − 211.·13-s − 246.·14-s − 715.·15-s + 256·16-s − 471.·17-s − 2.30e3·18-s + 2.70e3·19-s − 400·20-s + 1.76e3·21-s + 196.·22-s + 529·23-s − 1.83e3·24-s + 625·25-s + 847.·26-s + 9.54e3·27-s + 984.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.83·3-s + 0.5·4-s − 0.447·5-s − 1.29·6-s + 0.474·7-s − 0.353·8-s + 2.37·9-s + 0.316·10-s − 0.122·11-s + 0.918·12-s − 0.347·13-s − 0.335·14-s − 0.821·15-s + 0.250·16-s − 0.395·17-s − 1.67·18-s + 1.72·19-s − 0.223·20-s + 0.871·21-s + 0.0867·22-s + 0.208·23-s − 0.649·24-s + 0.200·25-s + 0.245·26-s + 2.51·27-s + 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.977583141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.977583141\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 + 25T \) |
| 23 | \( 1 - 529T \) |
| good | 3 | \( 1 - 28.6T + 243T^{2} \) |
| 7 | \( 1 - 61.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 49.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 211.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 471.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.70e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.38e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 771.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.08e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.49e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.34e5T + 8.58e9T^{2} \) |
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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
−11.20265948035100937901444414919, −9.912330162066025342140579739898, −9.328725491287333683688866319800, −8.236798548268365414097174748458, −7.82408109293918304386527350831, −6.84168730923530331892872229849, −4.77089812564201997999810399282, −3.40837861224487728673503283155, −2.45971105646415298079360477875, −1.14105596607225102079693092398,
1.14105596607225102079693092398, 2.45971105646415298079360477875, 3.40837861224487728673503283155, 4.77089812564201997999810399282, 6.84168730923530331892872229849, 7.82408109293918304386527350831, 8.236798548268365414097174748458, 9.328725491287333683688866319800, 9.912330162066025342140579739898, 11.20265948035100937901444414919
