| L(s) = 1 | + 32·2-s − 769.·3-s + 1.02e3·4-s + 86.6·5-s − 2.46e4·6-s + 1.68e4·7-s + 3.27e4·8-s + 4.14e5·9-s + 2.77e3·10-s + 9.01e5·11-s − 7.87e5·12-s − 9.73e5·13-s + 5.37e5·14-s − 6.66e4·15-s + 1.04e6·16-s + 2.11e6·17-s + 1.32e7·18-s + 1.29e7·19-s + 8.87e4·20-s − 1.29e7·21-s + 2.88e7·22-s − 8.14e6·23-s − 2.52e7·24-s − 4.88e7·25-s − 3.11e7·26-s − 1.82e8·27-s + 1.72e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.82·3-s + 0.5·4-s + 0.0124·5-s − 1.29·6-s + 0.377·7-s + 0.353·8-s + 2.33·9-s + 0.00876·10-s + 1.68·11-s − 0.913·12-s − 0.727·13-s + 0.267·14-s − 0.0226·15-s + 0.250·16-s + 0.360·17-s + 1.65·18-s + 1.20·19-s + 0.00620·20-s − 0.690·21-s + 1.19·22-s − 0.263·23-s − 0.646·24-s − 0.999·25-s − 0.514·26-s − 2.44·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.690975950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.690975950\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 3 | \( 1 + 769.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 86.6T + 4.88e7T^{2} \) |
| 11 | \( 1 - 9.01e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.73e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 2.11e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.29e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 8.14e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.30e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.44e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.16e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 4.50e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.65e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 3.38e7T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.41e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 2.83e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 8.40e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.00e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.08e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 8.59e8T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.43e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.42e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.02e11T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.75e10T + 7.15e21T^{2} \) |
| 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
−16.94825066007944380774289056864, −15.71501927515019861704860858257, −14.03100324301213769015968542094, −12.05593604904091829676104437355, −11.73486905984161137152504622273, −10.07209392008013572309450726942, −7.00760001551793928580440999491, −5.72519911583653890870624961928, −4.35572500071537038218130503238, −1.12253673883669893719518088295,
1.12253673883669893719518088295, 4.35572500071537038218130503238, 5.72519911583653890870624961928, 7.00760001551793928580440999491, 10.07209392008013572309450726942, 11.73486905984161137152504622273, 12.05593604904091829676104437355, 14.03100324301213769015968542094, 15.71501927515019861704860858257, 16.94825066007944380774289056864
