| L(s) = 1 | − 32·2-s + 1.02e3·4-s − 1.23e4·5-s + 1.68e4·7-s − 3.27e4·8-s + 3.93e5·10-s + 3.52e5·11-s + 1.58e6·13-s − 5.37e5·14-s + 1.04e6·16-s + 2.48e6·17-s − 1.61e7·19-s − 1.25e7·20-s − 1.12e7·22-s − 4.52e7·23-s + 1.02e8·25-s − 5.05e7·26-s + 1.72e7·28-s − 1.70e8·29-s − 1.42e7·31-s − 3.35e7·32-s − 7.96e7·34-s − 2.06e8·35-s + 1.89e8·37-s + 5.16e8·38-s + 4.03e8·40-s − 2.36e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.76·5-s + 0.377·7-s − 0.353·8-s + 1.24·10-s + 0.660·11-s + 1.18·13-s − 0.267·14-s + 0.250·16-s + 0.425·17-s − 1.49·19-s − 0.880·20-s − 0.466·22-s − 1.46·23-s + 2.09·25-s − 0.834·26-s + 0.188·28-s − 1.54·29-s − 0.0891·31-s − 0.176·32-s − 0.300·34-s − 0.665·35-s + 0.450·37-s + 1.05·38-s + 0.622·40-s − 0.318·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.7628366852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7628366852\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 5 | \( 1 + 1.23e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 3.52e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.58e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 2.48e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.61e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.52e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.70e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.42e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.89e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.36e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.90e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.17e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 6.05e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.82e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.02e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.64e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 6.23e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.55e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.48e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.22e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.88e10T + 7.15e21T^{2} \) |
| 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
−11.36192378720326327065893876188, −10.35503179964620310258406421635, −8.798131232932924542043476124657, −8.205460521680961862837593252827, −7.29290783230493898140955539923, −6.08533540971293932556919190863, −4.24662708513032283867612827320, −3.54176663593475208783971251477, −1.75375528676786622983506989363, −0.47683205405045735908427377916,
0.47683205405045735908427377916, 1.75375528676786622983506989363, 3.54176663593475208783971251477, 4.24662708513032283867612827320, 6.08533540971293932556919190863, 7.29290783230493898140955539923, 8.205460521680961862837593252827, 8.798131232932924542043476124657, 10.35503179964620310258406421635, 11.36192378720326327065893876188
