| L(s) = 1 | + 21.5·2-s + 27·3-s + 335.·4-s + 54.7·5-s + 581.·6-s − 565.·7-s + 4.47e3·8-s + 729·9-s + 1.17e3·10-s − 890.·11-s + 9.06e3·12-s + 1.24e4·13-s − 1.21e4·14-s + 1.47e3·15-s + 5.34e4·16-s − 2.69e3·17-s + 1.57e4·18-s − 5.60e4·19-s + 1.83e4·20-s − 1.52e4·21-s − 1.91e4·22-s + 1.21e4·23-s + 1.20e5·24-s − 7.51e4·25-s + 2.67e5·26-s + 1.96e4·27-s − 1.89e5·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 0.577·3-s + 2.62·4-s + 0.195·5-s + 1.09·6-s − 0.622·7-s + 3.09·8-s + 0.333·9-s + 0.372·10-s − 0.201·11-s + 1.51·12-s + 1.56·13-s − 1.18·14-s + 0.113·15-s + 3.26·16-s − 0.132·17-s + 0.634·18-s − 1.87·19-s + 0.513·20-s − 0.359·21-s − 0.383·22-s + 0.208·23-s + 1.78·24-s − 0.961·25-s + 2.98·26-s + 0.192·27-s − 1.63·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.372527103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.372527103\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 21.5T + 128T^{2} \) |
| 5 | \( 1 - 54.7T + 7.81e4T^{2} \) |
| 7 | \( 1 + 565.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 890.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.24e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.69e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.60e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 7.26e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.60e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.55e3T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.74e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.63e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.29e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.20e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.81e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 6.49e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.07e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.40e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.28e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.57e6T + 8.07e13T^{2} \) |
| 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
−13.32246753924745527850588487321, −12.72969970967303028600643993609, −11.38064006886655798372953793395, −10.29569796167023940317568120466, −8.401589609885091478693690626346, −6.74989734378973805663875310998, −5.91040683742320675326419691076, −4.29109766510332481824571749337, −3.29516518957632270922038236763, −1.93978661155807012447190946587,
1.93978661155807012447190946587, 3.29516518957632270922038236763, 4.29109766510332481824571749337, 5.91040683742320675326419691076, 6.74989734378973805663875310998, 8.401589609885091478693690626346, 10.29569796167023940317568120466, 11.38064006886655798372953793395, 12.72969970967303028600643993609, 13.32246753924745527850588487321
