| L(s) = 1 | − 4.57·2-s + 12.9·4-s + 5·5-s − 20.1·7-s − 22.4·8-s − 22.8·10-s + 66.3·11-s − 46.8·13-s + 91.9·14-s − 0.475·16-s − 47.6·17-s − 9.95·19-s + 64.5·20-s − 303.·22-s + 9.59·23-s + 25·25-s + 214.·26-s − 259.·28-s − 178.·29-s + 154.·31-s + 182.·32-s + 217.·34-s − 100.·35-s + 248.·37-s + 45.5·38-s − 112.·40-s + 249.·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.61·4-s + 0.447·5-s − 1.08·7-s − 0.993·8-s − 0.723·10-s + 1.81·11-s − 0.998·13-s + 1.75·14-s − 0.00743·16-s − 0.679·17-s − 0.120·19-s + 0.722·20-s − 2.94·22-s + 0.0869·23-s + 0.200·25-s + 1.61·26-s − 1.75·28-s − 1.14·29-s + 0.892·31-s + 1.00·32-s + 1.09·34-s − 0.485·35-s + 1.10·37-s + 0.194·38-s − 0.444·40-s + 0.951·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 4.57T + 8T^{2} \) |
| 7 | \( 1 + 20.1T + 343T^{2} \) |
| 11 | \( 1 - 66.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 9.95T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.59T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 248.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 212.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 475.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 546.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 419.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 545.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 447.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 358.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 651.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 813.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 200.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 252.T + 9.12e5T^{2} \) |
| 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
−9.983998711619038447682764097610, −9.353104879237615558517271498693, −9.007920393603724184827094486240, −7.66107425279408472166484796289, −6.74806795507664424535872186321, −6.15351137774435862970517732417, −4.28921387377713224539709325720, −2.70539323450455699494528846291, −1.40419267616158714528906795252, 0,
1.40419267616158714528906795252, 2.70539323450455699494528846291, 4.28921387377713224539709325720, 6.15351137774435862970517732417, 6.74806795507664424535872186321, 7.66107425279408472166484796289, 9.007920393603724184827094486240, 9.353104879237615558517271498693, 9.983998711619038447682764097610
