| L(s) = 1 | + 7.52·2-s + 9·3-s + 24.6·4-s − 25·5-s + 67.7·6-s − 234.·7-s − 55.3·8-s + 81·9-s − 188.·10-s + 121·11-s + 221.·12-s + 236.·13-s − 1.76e3·14-s − 225·15-s − 1.20e3·16-s − 608.·17-s + 609.·18-s − 1.79e3·19-s − 616.·20-s − 2.10e3·21-s + 910.·22-s − 4.77e3·23-s − 498.·24-s + 625·25-s + 1.77e3·26-s + 729·27-s − 5.76e3·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.770·4-s − 0.447·5-s + 0.768·6-s − 1.80·7-s − 0.305·8-s + 0.333·9-s − 0.594·10-s + 0.301·11-s + 0.444·12-s + 0.387·13-s − 2.40·14-s − 0.258·15-s − 1.17·16-s − 0.510·17-s + 0.443·18-s − 1.14·19-s − 0.344·20-s − 1.04·21-s + 0.401·22-s − 1.88·23-s − 0.176·24-s + 0.200·25-s + 0.515·26-s + 0.192·27-s − 1.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 7.52T + 32T^{2} \) |
| 7 | \( 1 + 234.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 236.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 608.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.38e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.45e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.10e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.34e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.83e5T + 8.58e9T^{2} \) |
| 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
−12.02877356937638650620117121053, −10.45728332715078813167465920634, −9.390463505813236589480313996394, −8.334723813427079207734442182447, −6.65570257910159012740639350911, −6.15650676064698743725429505115, −4.36323270705408538979636450099, −3.60938486706975797715607150314, −2.56966967452185228844934593381, 0,
2.56966967452185228844934593381, 3.60938486706975797715607150314, 4.36323270705408538979636450099, 6.15650676064698743725429505115, 6.65570257910159012740639350911, 8.334723813427079207734442182447, 9.390463505813236589480313996394, 10.45728332715078813167465920634, 12.02877356937638650620117121053
