| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 25·5-s + 36·6-s + 234.·7-s + 64·8-s + 81·9-s − 100·10-s − 283.·11-s + 144·12-s + 571.·13-s + 936.·14-s − 225·15-s + 256·16-s − 1.14e3·17-s + 324·18-s + 361·19-s − 400·20-s + 2.10e3·21-s − 1.13e3·22-s + 1.76e3·23-s + 576·24-s + 625·25-s + 2.28e3·26-s + 729·27-s + 3.74e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.80·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.705·11-s + 0.288·12-s + 0.937·13-s + 1.27·14-s − 0.258·15-s + 0.250·16-s − 0.958·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 1.04·21-s − 0.498·22-s + 0.696·23-s + 0.204·24-s + 0.200·25-s + 0.663·26-s + 0.192·27-s + 0.902·28-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
| \(L(3)\) |
\(\approx\) |
\(5.413455877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.413455877\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 234.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 283.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 571.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 129.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.57e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.95e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.87e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.42e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.69e5T + 8.58e9T^{2} \) |
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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
−10.16427704457563332727982036925, −8.677761318411738748358461628311, −8.202819986826285795526002296489, −7.41525819273288133749456226667, −6.28351099633870841229521982879, −4.91364004490220030302615037669, −4.52706095172651247982682527628, −3.26315407750506268718348924198, −2.15713015307528350536467426168, −1.08009196722127649610480725093,
1.08009196722127649610480725093, 2.15713015307528350536467426168, 3.26315407750506268718348924198, 4.52706095172651247982682527628, 4.91364004490220030302615037669, 6.28351099633870841229521982879, 7.41525819273288133749456226667, 8.202819986826285795526002296489, 8.677761318411738748358461628311, 10.16427704457563332727982036925
