| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 6·5-s − 36·6-s − 49·7-s + 64·8-s + 81·9-s − 24·10-s + 121·11-s − 144·12-s + 46·13-s − 196·14-s + 54·15-s + 256·16-s − 170·17-s + 324·18-s − 1.22e3·19-s − 96·20-s + 441·21-s + 484·22-s + 2.29e3·23-s − 576·24-s − 3.08e3·25-s + 184·26-s − 729·27-s − 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.107·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.0758·10-s + 0.301·11-s − 0.288·12-s + 0.0754·13-s − 0.267·14-s + 0.0619·15-s + 1/4·16-s − 0.142·17-s + 0.235·18-s − 0.777·19-s − 0.0536·20-s + 0.218·21-s + 0.213·22-s + 0.905·23-s − 0.204·24-s − 0.988·25-s + 0.0533·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 6 T + p^{5} T^{2} \) |
| 13 | \( 1 - 46 T + p^{5} T^{2} \) |
| 17 | \( 1 + 10 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 1224 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2296 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1154 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9636 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8322 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3246 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10652 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2860 T + p^{5} T^{2} \) |
| 53 | \( 1 + 28554 T + p^{5} T^{2} \) |
| 59 | \( 1 + 1300 T + p^{5} T^{2} \) |
| 61 | \( 1 + 30210 T + p^{5} T^{2} \) |
| 67 | \( 1 + 67228 T + p^{5} T^{2} \) |
| 71 | \( 1 + 45648 T + p^{5} T^{2} \) |
| 73 | \( 1 - 21390 T + p^{5} T^{2} \) |
| 79 | \( 1 + 8184 T + p^{5} T^{2} \) |
| 83 | \( 1 + 47048 T + p^{5} T^{2} \) |
| 89 | \( 1 - 126890 T + p^{5} T^{2} \) |
| 97 | \( 1 + 46718 T + p^{5} T^{2} \) |
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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
−10.04035470545601234513479943046, −8.939598785607062235490696764510, −7.76606526558102809946716709565, −6.68931919778189081875893678829, −6.08597435181725536914695940775, −4.95567382243921588986916298659, −4.07197499268209730800499848245, −2.91059454941126043330106154688, −1.49252387936569761533650960879, 0,
1.49252387936569761533650960879, 2.91059454941126043330106154688, 4.07197499268209730800499848245, 4.95567382243921588986916298659, 6.08597435181725536914695940775, 6.68931919778189081875893678829, 7.76606526558102809946716709565, 8.939598785607062235490696764510, 10.04035470545601234513479943046
