| L(s) = 1 | − 1.06·2-s − 0.862·4-s + 1.22·5-s + 1.53·7-s + 3.05·8-s − 1.30·10-s − 2.34·13-s − 1.63·14-s − 1.53·16-s − 3.55·17-s + 2.86·19-s − 1.05·20-s − 2.29·23-s − 3.49·25-s + 2.49·26-s − 1.32·28-s + 8.29·29-s + 4.83·31-s − 4.47·32-s + 3.78·34-s + 1.88·35-s + 4.02·37-s − 3.05·38-s + 3.74·40-s + 4.84·41-s + 7.52·43-s + 2.44·46-s + ⋯ |
| L(s) = 1 | − 0.754·2-s − 0.431·4-s + 0.549·5-s + 0.579·7-s + 1.07·8-s − 0.414·10-s − 0.649·13-s − 0.436·14-s − 0.383·16-s − 0.861·17-s + 0.656·19-s − 0.236·20-s − 0.478·23-s − 0.698·25-s + 0.489·26-s − 0.249·28-s + 1.53·29-s + 0.868·31-s − 0.790·32-s + 0.649·34-s + 0.318·35-s + 0.661·37-s − 0.495·38-s + 0.592·40-s + 0.755·41-s + 1.14·43-s + 0.360·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182256290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182256290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.06T + 2T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 29 | \( 1 - 8.29T + 29T^{2} \) |
| 31 | \( 1 - 4.83T + 31T^{2} \) |
| 37 | \( 1 - 4.02T + 37T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 - 7.52T + 43T^{2} \) |
| 47 | \( 1 - 9.15T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 0.613T + 61T^{2} \) |
| 67 | \( 1 - 8.65T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 1.94T + 79T^{2} \) |
| 83 | \( 1 + 5.33T + 83T^{2} \) |
| 89 | \( 1 + 5.84T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.676076325151380445769760534050, −7.981858174958317609995090906514, −7.45102202839965116221995558942, −6.45756609371745694996584714956, −5.61481537943390231219753659820, −4.68235738848426049233606672907, −4.24842512548843240314288587100, −2.77949779672713361832230043415, −1.84709345143478627027180561384, −0.75266602401543249156380809072,
0.75266602401543249156380809072, 1.84709345143478627027180561384, 2.77949779672713361832230043415, 4.24842512548843240314288587100, 4.68235738848426049233606672907, 5.61481537943390231219753659820, 6.45756609371745694996584714956, 7.45102202839965116221995558942, 7.981858174958317609995090906514, 8.676076325151380445769760534050
