| L(s) = 1 | − 8.05·2-s − 20.7·3-s + 32.9·4-s − 25·5-s + 167.·6-s − 117.·7-s − 7.50·8-s + 189.·9-s + 201.·10-s − 684.·12-s + 58.4·13-s + 947.·14-s + 519.·15-s − 993.·16-s − 695.·17-s − 1.52e3·18-s − 1.37e3·19-s − 823.·20-s + 2.44e3·21-s + 44.1·23-s + 156.·24-s + 625·25-s − 470.·26-s + 1.11e3·27-s − 3.87e3·28-s − 4.71e3·29-s − 4.19e3·30-s + ⋯ |
| L(s) = 1 | − 1.42·2-s − 1.33·3-s + 1.02·4-s − 0.447·5-s + 1.90·6-s − 0.906·7-s − 0.0414·8-s + 0.780·9-s + 0.637·10-s − 1.37·12-s + 0.0958·13-s + 1.29·14-s + 0.596·15-s − 0.970·16-s − 0.583·17-s − 1.11·18-s − 0.875·19-s − 0.460·20-s + 1.21·21-s + 0.0174·23-s + 0.0553·24-s + 0.200·25-s − 0.136·26-s + 0.293·27-s − 0.933·28-s − 1.04·29-s − 0.850·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 8.05T + 32T^{2} \) |
| 3 | \( 1 + 20.7T + 243T^{2} \) |
| 7 | \( 1 + 117.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 58.4T + 3.71e5T^{2} \) |
| 17 | \( 1 + 695.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.37e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 44.1T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.48e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.36e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.19e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.36e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.33e4T + 8.58e9T^{2} \) |
| 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
−9.486280191008810199307052523813, −8.729592851548824922609880802478, −7.67370976536976368401679867787, −6.80056642827092212410114485576, −6.15229453265624160169051999486, −4.96111356798002675092166736963, −3.76190964175054584303849667963, −2.08527970968633318816483718325, −0.67302673153073575219010226191, 0,
0.67302673153073575219010226191, 2.08527970968633318816483718325, 3.76190964175054584303849667963, 4.96111356798002675092166736963, 6.15229453265624160169051999486, 6.80056642827092212410114485576, 7.67370976536976368401679867787, 8.729592851548824922609880802478, 9.486280191008810199307052523813
