| L(s) = 1 | − 7.34·2-s + 21.4·3-s + 21.9·4-s + 25·5-s − 157.·6-s + 84.2·7-s + 73.5·8-s + 216.·9-s − 183.·10-s + 121·11-s + 471.·12-s − 947.·13-s − 618.·14-s + 535.·15-s − 1.24e3·16-s + 938.·17-s − 1.58e3·18-s − 361·19-s + 549.·20-s + 1.80e3·21-s − 889.·22-s − 752.·23-s + 1.57e3·24-s + 625·25-s + 6.96e3·26-s − 571.·27-s + 1.85e3·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 1.37·3-s + 0.687·4-s + 0.447·5-s − 1.78·6-s + 0.649·7-s + 0.406·8-s + 0.890·9-s − 0.580·10-s + 0.301·11-s + 0.944·12-s − 1.55·13-s − 0.843·14-s + 0.614·15-s − 1.21·16-s + 0.787·17-s − 1.15·18-s − 0.229·19-s + 0.307·20-s + 0.893·21-s − 0.391·22-s − 0.296·23-s + 0.558·24-s + 0.200·25-s + 2.01·26-s − 0.150·27-s + 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 7.34T + 32T^{2} \) |
| 3 | \( 1 - 21.4T + 243T^{2} \) |
| 7 | \( 1 - 84.2T + 1.68e4T^{2} \) |
| 13 | \( 1 + 947.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 938.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 752.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.46e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.70e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.73e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 618.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.36e4T + 8.58e9T^{2} \) |
| 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
−8.934749058752884084572033737284, −7.921708164613313411487294476280, −7.71669163645982366107201789086, −6.71590330189124135588041020396, −5.24486641971307999421876695508, −4.29252216160167035824500726287, −2.98609090278967316940436996247, −2.08455547203231064394672929489, −1.39333023907185178038667914520, 0,
1.39333023907185178038667914520, 2.08455547203231064394672929489, 2.98609090278967316940436996247, 4.29252216160167035824500726287, 5.24486641971307999421876695508, 6.71590330189124135588041020396, 7.71669163645982366107201789086, 7.921708164613313411487294476280, 8.934749058752884084572033737284
