| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s + 22.0·7-s − 64·8-s + 81·9-s + 100·10-s + 191.·11-s + 144·12-s − 381.·13-s − 88.3·14-s − 225·15-s + 256·16-s − 92.9·17-s − 324·18-s + 361·19-s − 400·20-s + 198.·21-s − 764.·22-s − 1.29e3·23-s − 576·24-s + 625·25-s + 1.52e3·26-s + 729·27-s + 353.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.170·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.476·11-s + 0.288·12-s − 0.626·13-s − 0.120·14-s − 0.258·15-s + 0.250·16-s − 0.0780·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.0983·21-s − 0.336·22-s − 0.509·23-s − 0.204·24-s + 0.200·25-s + 0.442·26-s + 0.192·27-s + 0.0851·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 22.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 191.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 381.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 92.9T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.24e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.67e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.26e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 804.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.76e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.29e5T + 8.58e9T^{2} \) |
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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
−9.407729114157246009189808362786, −8.682223946106892279775396287594, −7.77528560038212757444496951169, −7.17636895198622305697647106857, −6.05895498985184078570145838090, −4.69865082642275553494312378484, −3.60632690222014088442786101341, −2.48397992505714044337003320751, −1.33882896458984638232808476385, 0,
1.33882896458984638232808476385, 2.48397992505714044337003320751, 3.60632690222014088442786101341, 4.69865082642275553494312378484, 6.05895498985184078570145838090, 7.17636895198622305697647106857, 7.77528560038212757444496951169, 8.682223946106892279775396287594, 9.407729114157246009189808362786
