L(s) = 1 | − 1.89·2-s + 9·3-s − 28.4·4-s − 17.0·6-s + 191.·7-s + 114.·8-s + 81·9-s − 121·11-s − 255.·12-s + 536.·13-s − 363.·14-s + 692.·16-s − 1.32e3·17-s − 153.·18-s − 912.·19-s + 1.72e3·21-s + 229.·22-s − 4.12e3·23-s + 1.02e3·24-s − 1.01e3·26-s + 729·27-s − 5.45e3·28-s + 340.·29-s + 4.69e3·31-s − 4.97e3·32-s − 1.08e3·33-s + 2.50e3·34-s + ⋯ |
L(s) = 1 | − 0.334·2-s + 0.577·3-s − 0.887·4-s − 0.193·6-s + 1.48·7-s + 0.632·8-s + 0.333·9-s − 0.301·11-s − 0.512·12-s + 0.881·13-s − 0.495·14-s + 0.676·16-s − 1.11·17-s − 0.111·18-s − 0.579·19-s + 0.855·21-s + 0.100·22-s − 1.62·23-s + 0.364·24-s − 0.295·26-s + 0.192·27-s − 1.31·28-s + 0.0751·29-s + 0.877·31-s − 0.858·32-s − 0.174·33-s + 0.372·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 + 1.89T + 32T^{2} \) |
| 7 | \( 1 - 191.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 536.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 912.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 340.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 628.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.66e5T + 8.58e9T^{2} \) |
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
−8.711780662570291899303641942281, −8.383141167099228397044259501314, −7.79877208630789189517554195389, −6.53901730745869826229878044288, −5.28142602999828001038395361400, −4.47198534724587368808503715675, −3.76211355090268241606048141649, −2.17237791311444111259987407087, −1.35405676618669954579554302406, 0,
1.35405676618669954579554302406, 2.17237791311444111259987407087, 3.76211355090268241606048141649, 4.47198534724587368808503715675, 5.28142602999828001038395361400, 6.53901730745869826229878044288, 7.79877208630789189517554195389, 8.383141167099228397044259501314, 8.711780662570291899303641942281