| L(s) = 1 | + 4·2-s − 2.70·3-s + 16·4-s + 25·5-s − 10.8·6-s − 158.·7-s + 64·8-s − 235.·9-s + 100·10-s + 280.·11-s − 43.3·12-s + 330.·13-s − 632.·14-s − 67.7·15-s + 256·16-s + 1.50e3·17-s − 942.·18-s + 1.60e3·19-s + 400·20-s + 428.·21-s + 1.12e3·22-s + 529·23-s − 173.·24-s + 625·25-s + 1.32e3·26-s + 1.29e3·27-s − 2.53e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.173·3-s + 0.5·4-s + 0.447·5-s − 0.122·6-s − 1.22·7-s + 0.353·8-s − 0.969·9-s + 0.316·10-s + 0.697·11-s − 0.0868·12-s + 0.542·13-s − 0.862·14-s − 0.0777·15-s + 0.250·16-s + 1.26·17-s − 0.685·18-s + 1.02·19-s + 0.223·20-s + 0.212·21-s + 0.493·22-s + 0.208·23-s − 0.0614·24-s + 0.200·25-s + 0.383·26-s + 0.342·27-s − 0.610·28-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
| \(L(3)\) |
\(\approx\) |
\(2.972975439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.972975439\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.70T + 243T^{2} \) |
| 7 | \( 1 + 158.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 280.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 330.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.50e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.60e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.78e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.59e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.34e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.11e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.89e4T + 8.58e9T^{2} \) |
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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
−11.67945658760804733218634260444, −10.36199824708435183442331992363, −9.563558767116775231241126330332, −8.405577375455636282925264337849, −6.92182809743135663410199707712, −6.09229428797931296122307380344, −5.30294291223524876694027448388, −3.64064138480563134386134311583, −2.82069548758850593552509454282, −0.968438749829031031849432693166,
0.968438749829031031849432693166, 2.82069548758850593552509454282, 3.64064138480563134386134311583, 5.30294291223524876694027448388, 6.09229428797931296122307380344, 6.92182809743135663410199707712, 8.405577375455636282925264337849, 9.563558767116775231241126330332, 10.36199824708435183442331992363, 11.67945658760804733218634260444
