| L(s) = 1 | − 9.25·2-s + 9·3-s + 53.6·4-s + 25·5-s − 83.2·6-s + 36.4·7-s − 200.·8-s + 81·9-s − 231.·10-s − 121·11-s + 482.·12-s − 878.·13-s − 337.·14-s + 225·15-s + 138.·16-s + 155.·17-s − 749.·18-s − 1.93e3·19-s + 1.34e3·20-s + 328.·21-s + 1.11e3·22-s + 1.92e3·23-s − 1.80e3·24-s + 625·25-s + 8.12e3·26-s + 729·27-s + 1.95e3·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 0.577·3-s + 1.67·4-s + 0.447·5-s − 0.944·6-s + 0.281·7-s − 1.10·8-s + 0.333·9-s − 0.731·10-s − 0.301·11-s + 0.968·12-s − 1.44·13-s − 0.459·14-s + 0.258·15-s + 0.135·16-s + 0.130·17-s − 0.545·18-s − 1.22·19-s + 0.749·20-s + 0.162·21-s + 0.493·22-s + 0.759·23-s − 0.639·24-s + 0.200·25-s + 2.35·26-s + 0.192·27-s + 0.471·28-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 | \( 1 + 9.25T + 32T^{2} \) |
| 7 | \( 1 - 36.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 878.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 155.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.93e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.92e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 480.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.89e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.47e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.14e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.67e4T + 8.58e9T^{2} \) |
| 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
−10.95234006513408669213722949422, −10.11555758995383618390600633519, −9.369676587058756600020413038646, −8.446637262097494723882953486691, −7.58765644839194753908363378024, −6.60430373285099731449965007460, −4.81813386704793197314484513086, −2.69442915745644190984397759167, −1.65207071322074254971014894369, 0,
1.65207071322074254971014894369, 2.69442915745644190984397759167, 4.81813386704793197314484513086, 6.60430373285099731449965007460, 7.58765644839194753908363378024, 8.446637262097494723882953486691, 9.369676587058756600020413038646, 10.11555758995383618390600633519, 10.95234006513408669213722949422
