| L(s) = 1 | − 4·2-s − 26.2·3-s + 16·4-s − 25·5-s + 105.·6-s + 152.·7-s − 64·8-s + 448.·9-s + 100·10-s + 632.·11-s − 420.·12-s − 657.·13-s − 610.·14-s + 657.·15-s + 256·16-s − 1.10e3·17-s − 1.79e3·18-s − 970.·19-s − 400·20-s − 4.01e3·21-s − 2.53e3·22-s + 529·23-s + 1.68e3·24-s + 625·25-s + 2.63e3·26-s − 5.40e3·27-s + 2.44e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.68·3-s + 0.5·4-s − 0.447·5-s + 1.19·6-s + 1.17·7-s − 0.353·8-s + 1.84·9-s + 0.316·10-s + 1.57·11-s − 0.843·12-s − 1.07·13-s − 0.832·14-s + 0.754·15-s + 0.250·16-s − 0.923·17-s − 1.30·18-s − 0.616·19-s − 0.223·20-s − 1.98·21-s − 1.11·22-s + 0.208·23-s + 0.596·24-s + 0.200·25-s + 0.763·26-s − 1.42·27-s + 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6846568986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6846568986\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 + 25T \) |
| 23 | \( 1 - 529T \) |
| good | 3 | \( 1 + 26.2T + 243T^{2} \) |
| 7 | \( 1 - 152.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 632.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 657.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 970.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 769.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.21e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.57e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.08e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.64e4T + 8.58e9T^{2} \) |
| 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
−11.34213815506731631486014344647, −10.69010863821201381877200098541, −9.518802312027121456772069156259, −8.366546019505757323434021902448, −7.10628979350392348552362219402, −6.46148087155750745744598313298, −5.10425690311495875114924774690, −4.24475267244651717872872205451, −1.79214050293872663002364285607, −0.61449721762948926976222998358,
0.61449721762948926976222998358, 1.79214050293872663002364285607, 4.24475267244651717872872205451, 5.10425690311495875114924774690, 6.46148087155750745744598313298, 7.10628979350392348552362219402, 8.366546019505757323434021902448, 9.518802312027121456772069156259, 10.69010863821201381877200098541, 11.34213815506731631486014344647
