| L(s) = 1 | − 1.73·2-s − 28.9·4-s + 91.0·5-s − 76.4·7-s + 106.·8-s − 158.·10-s − 476.·11-s − 1.15e3·13-s + 132.·14-s + 742.·16-s − 1.28e3·17-s + 432.·19-s − 2.63e3·20-s + 828.·22-s − 2.38e3·23-s + 5.17e3·25-s + 2.00e3·26-s + 2.21e3·28-s − 972.·29-s − 3.42e3·31-s − 4.68e3·32-s + 2.23e3·34-s − 6.95e3·35-s + 7.54e3·37-s − 752.·38-s + 9.66e3·40-s + 7.09e3·41-s + ⋯ |
| L(s) = 1 | − 0.307·2-s − 0.905·4-s + 1.62·5-s − 0.589·7-s + 0.585·8-s − 0.501·10-s − 1.18·11-s − 1.89·13-s + 0.181·14-s + 0.725·16-s − 1.07·17-s + 0.274·19-s − 1.47·20-s + 0.364·22-s − 0.939·23-s + 1.65·25-s + 0.581·26-s + 0.533·28-s − 0.214·29-s − 0.640·31-s − 0.808·32-s + 0.331·34-s − 0.960·35-s + 0.906·37-s − 0.0845·38-s + 0.954·40-s + 0.659·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9308031084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9308031084\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 32T^{2} \) |
| 5 | \( 1 - 91.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 76.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 476.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.28e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 432.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 972.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.54e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.72e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.84e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.37e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 193.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.86e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.94e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.49e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.74e5T + 8.58e9T^{2} \) |
| 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
−9.655374220394290331123639728312, −9.142292030091939194133977021411, −7.968832703908913766332161767537, −7.08501421943143238322636674066, −5.88842350673084316800601884942, −5.25029096369924904051411604420, −4.37208904379669477560222013254, −2.71711177374456866571012993800, −2.03798900248305287499773427328, −0.44946478412022155893551519986,
0.44946478412022155893551519986, 2.03798900248305287499773427328, 2.71711177374456866571012993800, 4.37208904379669477560222013254, 5.25029096369924904051411604420, 5.88842350673084316800601884942, 7.08501421943143238322636674066, 7.968832703908913766332161767537, 9.142292030091939194133977021411, 9.655374220394290331123639728312
