| L(s) = 1 | − 8·2-s + 20.4·3-s + 64·4-s + 375.·5-s − 163.·6-s − 980.·7-s − 512·8-s − 1.76e3·9-s − 3.00e3·10-s − 3.37e3·11-s + 1.31e3·12-s + 3.43e3·13-s + 7.84e3·14-s + 7.69e3·15-s + 4.09e3·16-s − 4.74e3·17-s + 1.41e4·18-s − 1.12e4·19-s + 2.40e4·20-s − 2.01e4·21-s + 2.70e4·22-s − 1.77e4·23-s − 1.04e4·24-s + 6.27e4·25-s − 2.74e4·26-s − 8.10e4·27-s − 6.27e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.438·3-s + 0.5·4-s + 1.34·5-s − 0.309·6-s − 1.08·7-s − 0.353·8-s − 0.807·9-s − 0.949·10-s − 0.764·11-s + 0.219·12-s + 0.433·13-s + 0.764·14-s + 0.588·15-s + 0.250·16-s − 0.234·17-s + 0.571·18-s − 0.376·19-s + 0.671·20-s − 0.473·21-s + 0.540·22-s − 0.303·23-s − 0.154·24-s + 0.802·25-s − 0.306·26-s − 0.792·27-s − 0.540·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 37 | \( 1 - 5.06e4T \) |
| good | 3 | \( 1 - 20.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 375.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 980.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.37e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.43e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.74e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.13e4T + 2.75e10T^{2} \) |
| 41 | \( 1 - 2.78e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.13e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.97e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.69e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.02e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.85e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.28e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.37e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.63e6T + 8.07e13T^{2} \) |
| 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
−12.84750714619757282843393535269, −11.15151075117957220856752637879, −9.989728201688939474581122945223, −9.301337930865519238212383203799, −8.190153499014499833802761633206, −6.55412350511213497195160477471, −5.63118640598797215313043541416, −3.12971676067467901245787159769, −1.99323057322219530211809715948, 0,
1.99323057322219530211809715948, 3.12971676067467901245787159769, 5.63118640598797215313043541416, 6.55412350511213497195160477471, 8.190153499014499833802761633206, 9.301337930865519238212383203799, 9.989728201688939474581122945223, 11.15151075117957220856752637879, 12.84750714619757282843393535269
