| L(s) = 1 | + 0.732·2-s − 1.46·4-s + 5-s − 4.73·7-s − 2.53·8-s + 0.732·10-s − 5.73·11-s + 1.46·13-s − 3.46·14-s + 1.07·16-s − 2.73·17-s + 4.46·19-s − 1.46·20-s − 4.19·22-s − 3.46·23-s + 25-s + 1.07·26-s + 6.92·28-s + 3.19·29-s − 3·31-s + 5.85·32-s − 2·34-s − 4.73·35-s − 2.73·37-s + 3.26·38-s − 2.53·40-s − 7.19·41-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 0.732·4-s + 0.447·5-s − 1.78·7-s − 0.896·8-s + 0.231·10-s − 1.72·11-s + 0.406·13-s − 0.925·14-s + 0.267·16-s − 0.662·17-s + 1.02·19-s − 0.327·20-s − 0.894·22-s − 0.722·23-s + 0.200·25-s + 0.210·26-s + 1.30·28-s + 0.593·29-s − 0.538·31-s + 1.03·32-s − 0.342·34-s − 0.799·35-s − 0.449·37-s + 0.530·38-s − 0.400·40-s − 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 2 | \( 1 - 0.732T + 2T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 - 0.196T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + 9.66T + 97T^{2} \) |
| 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
−10.53395455354729580116053469386, −9.891659987097994186587409226244, −9.160920829589346233820277538329, −8.113279120425039489990623479181, −6.77562535273111037724955155460, −5.84110997200682682606964101848, −5.05003835867806828695676183738, −3.61864026751503349114017566624, −2.75549156342552231674726398882, 0,
2.75549156342552231674726398882, 3.61864026751503349114017566624, 5.05003835867806828695676183738, 5.84110997200682682606964101848, 6.77562535273111037724955155460, 8.113279120425039489990623479181, 9.160920829589346233820277538329, 9.891659987097994186587409226244, 10.53395455354729580116053469386
