| L(s) = 1 | + 9.71·2-s + 9·3-s + 62.3·4-s − 24.4·5-s + 87.4·6-s − 49·7-s + 294.·8-s + 81·9-s − 237.·10-s − 694.·11-s + 561.·12-s − 607.·13-s − 475.·14-s − 219.·15-s + 868.·16-s + 24.5·17-s + 786.·18-s − 2.22e3·19-s − 1.52e3·20-s − 441·21-s − 6.74e3·22-s + 529·23-s + 2.65e3·24-s − 2.52e3·25-s − 5.89e3·26-s + 729·27-s − 3.05e3·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.94·4-s − 0.436·5-s + 0.991·6-s − 0.377·7-s + 1.62·8-s + 0.333·9-s − 0.750·10-s − 1.73·11-s + 1.12·12-s − 0.996·13-s − 0.648·14-s − 0.252·15-s + 0.847·16-s + 0.0206·17-s + 0.572·18-s − 1.41·19-s − 0.851·20-s − 0.218·21-s − 2.97·22-s + 0.208·23-s + 0.940·24-s − 0.809·25-s − 1.71·26-s + 0.192·27-s − 0.736·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 9T \) |
| 7 | \( 1 + 49T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 9.71T + 32T^{2} \) |
| 5 | \( 1 + 24.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 694.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 607.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 24.5T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.22e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.85e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.46e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.06e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.72e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.00e5T + 8.58e9T^{2} \) |
| 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
−10.10231286077763991518752732533, −8.613027694601164889795708469162, −7.63685073505326638088652104553, −6.85250865284648331410796856709, −5.70577688736960887124391797050, −4.79950259287290902621120353934, −3.98375062192874446187638675946, −2.83035975309459679926479131352, −2.28341825021582844320464626912, 0,
2.28341825021582844320464626912, 2.83035975309459679926479131352, 3.98375062192874446187638675946, 4.79950259287290902621120353934, 5.70577688736960887124391797050, 6.85250865284648331410796856709, 7.63685073505326638088652104553, 8.613027694601164889795708469162, 10.10231286077763991518752732533
