| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 3.07·5-s + 6·6-s + 8.18·7-s − 8·8-s + 9·9-s − 6.14·10-s + 11·11-s − 12·12-s − 51.2·13-s − 16.3·14-s − 9.22·15-s + 16·16-s − 38.5·17-s − 18·18-s + 107.·19-s + 12.2·20-s − 24.5·21-s − 22·22-s + 41.5·23-s + 24·24-s − 115.·25-s + 102.·26-s − 27·27-s + 32.7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.274·5-s + 0.408·6-s + 0.442·7-s − 0.353·8-s + 0.333·9-s − 0.194·10-s + 0.301·11-s − 0.288·12-s − 1.09·13-s − 0.312·14-s − 0.158·15-s + 0.250·16-s − 0.549·17-s − 0.235·18-s + 1.29·19-s + 0.137·20-s − 0.255·21-s − 0.213·22-s + 0.377·23-s + 0.204·24-s − 0.924·25-s + 0.772·26-s − 0.192·27-s + 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2442 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2442 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 37 | \( 1 + 37T \) |
| good | 5 | \( 1 - 3.07T + 125T^{2} \) |
| 7 | \( 1 - 8.18T + 343T^{2} \) |
| 13 | \( 1 + 51.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 38.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 182.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 142.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 452.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 194.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 520.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 72.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 284.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 433.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 262.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 388.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 249.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 19.5T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.163683742502342339292929199256, −7.42760947516198417364217204092, −6.83656888659223446267736548431, −5.91567335784536672834909477132, −5.16233908476653303013974024278, −4.35978695420211439573414477985, −3.07576215194056819329354427960, −2.04425385286698435407081551166, −1.10765722069248408030823971402, 0,
1.10765722069248408030823971402, 2.04425385286698435407081551166, 3.07576215194056819329354427960, 4.35978695420211439573414477985, 5.16233908476653303013974024278, 5.91567335784536672834909477132, 6.83656888659223446267736548431, 7.42760947516198417364217204092, 8.163683742502342339292929199256
