| L(s) = 1 | + 5-s − 3·7-s + 5·11-s − 4·13-s − 8·17-s + 2·19-s − 2·23-s − 4·25-s − 6·29-s + 7·31-s − 3·35-s + 6·37-s − 6·41-s − 2·43-s − 6·47-s + 2·49-s − 5·53-s + 5·55-s − 4·59-s + 8·61-s − 4·65-s − 10·67-s + 8·71-s + 73-s − 15·77-s − 16·79-s − 11·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 1.50·11-s − 1.10·13-s − 1.94·17-s + 0.458·19-s − 0.417·23-s − 4/5·25-s − 1.11·29-s + 1.25·31-s − 0.507·35-s + 0.986·37-s − 0.937·41-s − 0.304·43-s − 0.875·47-s + 2/7·49-s − 0.686·53-s + 0.674·55-s − 0.520·59-s + 1.02·61-s − 0.496·65-s − 1.22·67-s + 0.949·71-s + 0.117·73-s − 1.70·77-s − 1.80·79-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−9.262617565466732478169009238742, −8.224236138838626236588097788287, −7.05717799238262722068269243873, −6.55960464539303925130968330201, −5.90728495331563131351841280691, −4.67787935460078313498836162326, −3.89814181993739315556342397740, −2.80179600302493653223083598111, −1.76559844837423464517024482336, 0,
1.76559844837423464517024482336, 2.80179600302493653223083598111, 3.89814181993739315556342397740, 4.67787935460078313498836162326, 5.90728495331563131351841280691, 6.55960464539303925130968330201, 7.05717799238262722068269243873, 8.224236138838626236588097788287, 9.262617565466732478169009238742
