| L(s) = 1 | − 1.54·2-s − 3·3-s − 5.62·4-s − 4.20·5-s + 4.62·6-s − 32.1·7-s + 20.9·8-s + 9·9-s + 6.48·10-s + 11·11-s + 16.8·12-s − 90.3·13-s + 49.5·14-s + 12.6·15-s + 12.6·16-s − 98.2·17-s − 13.8·18-s + 48.8·19-s + 23.6·20-s + 96.3·21-s − 16.9·22-s + 21.6·23-s − 62.9·24-s − 107.·25-s + 139.·26-s − 27·27-s + 180.·28-s + ⋯ |
| L(s) = 1 | − 0.544·2-s − 0.577·3-s − 0.703·4-s − 0.376·5-s + 0.314·6-s − 1.73·7-s + 0.928·8-s + 0.333·9-s + 0.205·10-s + 0.301·11-s + 0.405·12-s − 1.92·13-s + 0.945·14-s + 0.217·15-s + 0.197·16-s − 1.40·17-s − 0.181·18-s + 0.589·19-s + 0.264·20-s + 1.00·21-s − 0.164·22-s + 0.196·23-s − 0.535·24-s − 0.858·25-s + 1.05·26-s − 0.192·27-s + 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 1.54T + 8T^{2} \) |
| 5 | \( 1 + 4.20T + 125T^{2} \) |
| 7 | \( 1 + 32.1T + 343T^{2} \) |
| 13 | \( 1 + 90.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 98.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 21.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 57.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 362.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 128.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 363.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 213.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 543.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 70.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 354.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 438.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 321.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 197.T + 9.12e5T^{2} \) |
| 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
−8.563575528278771955486056539032, −7.53020799751184028078786749572, −6.90047432556690418288389739101, −6.21196233819478805862931986653, −5.03249665325030023370465516082, −4.43510021466244743462547675978, −3.43937097656596640583487122858, −2.34085976241641907874462702536, −0.66690969413702292024890482407, 0,
0.66690969413702292024890482407, 2.34085976241641907874462702536, 3.43937097656596640583487122858, 4.43510021466244743462547675978, 5.03249665325030023370465516082, 6.21196233819478805862931986653, 6.90047432556690418288389739101, 7.53020799751184028078786749572, 8.563575528278771955486056539032
