| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s + 5·11-s + 4·13-s − 4·16-s − 4·17-s − 5·19-s + 2·20-s + 10·22-s + 6·23-s + 25-s + 8·26-s − 5·29-s − 9·31-s − 8·32-s − 8·34-s − 10·37-s − 10·38-s + 7·41-s − 2·43-s + 10·44-s + 12·46-s + 2·47-s − 7·49-s + 2·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s + 1.50·11-s + 1.10·13-s − 16-s − 0.970·17-s − 1.14·19-s + 0.447·20-s + 2.13·22-s + 1.25·23-s + 1/5·25-s + 1.56·26-s − 0.928·29-s − 1.61·31-s − 1.41·32-s − 1.37·34-s − 1.64·37-s − 1.62·38-s + 1.09·41-s − 0.304·43-s + 1.50·44-s + 1.76·46-s + 0.291·47-s − 49-s + 0.282·50-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
| \(L(1)\) |
\(\approx\) |
\(2.957866634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957866634\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−11.31548780266267451963425405875, −10.84103814713817700544872039181, −9.161742231918082706521428553592, −8.824991276786141939896068673100, −6.94257636940465251066789252595, −6.35526976205125359438346080603, −5.41659698665533312416876873683, −4.23466268327432431147155666300, −3.49410387761142976656546809271, −1.89799498723585767421033230965,
1.89799498723585767421033230965, 3.49410387761142976656546809271, 4.23466268327432431147155666300, 5.41659698665533312416876873683, 6.35526976205125359438346080603, 6.94257636940465251066789252595, 8.824991276786141939896068673100, 9.161742231918082706521428553592, 10.84103814713817700544872039181, 11.31548780266267451963425405875
