| L(s) = 1 | − 5.78·2-s + 9·3-s + 1.45·4-s − 52.0·6-s − 17.6·7-s + 176.·8-s + 81·9-s + 121·11-s + 13.1·12-s − 674.·13-s + 102.·14-s − 1.06e3·16-s + 2.11e3·17-s − 468.·18-s − 2.30e3·19-s − 158.·21-s − 699.·22-s + 3.07e3·23-s + 1.59e3·24-s + 3.90e3·26-s + 729·27-s − 25.7·28-s − 1.43e3·29-s − 5.15e3·31-s + 526.·32-s + 1.08e3·33-s − 1.22e4·34-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.577·3-s + 0.0455·4-s − 0.590·6-s − 0.136·7-s + 0.975·8-s + 0.333·9-s + 0.301·11-s + 0.0262·12-s − 1.10·13-s + 0.139·14-s − 1.04·16-s + 1.77·17-s − 0.340·18-s − 1.46·19-s − 0.0786·21-s − 0.308·22-s + 1.21·23-s + 0.563·24-s + 1.13·26-s + 0.192·27-s − 0.00619·28-s − 0.317·29-s − 0.963·31-s + 0.0909·32-s + 0.174·33-s − 1.81·34-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 | \( 1 + 5.78T + 32T^{2} \) |
| 7 | \( 1 + 17.6T + 1.68e4T^{2} \) |
| 13 | \( 1 + 674.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.11e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.30e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.45e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 378.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.52e5T + 8.58e9T^{2} \) |
| 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
−9.090023742815341498142174035297, −8.336691385767081788545429421265, −7.52952586048962628361150951615, −6.91401248555190957101366837079, −5.48504101154572855845718291019, −4.48601481583481356968311315583, −3.41723391003108686861821456561, −2.19454320605859059652973046126, −1.14705849442005581393992145520, 0,
1.14705849442005581393992145520, 2.19454320605859059652973046126, 3.41723391003108686861821456561, 4.48601481583481356968311315583, 5.48504101154572855845718291019, 6.91401248555190957101366837079, 7.52952586048962628361150951615, 8.336691385767081788545429421265, 9.090023742815341498142174035297
