| L(s) = 1 | + 5-s + 7-s − 4·13-s + 6·17-s + 2·19-s + 6·23-s + 25-s + 2·31-s + 35-s + 2·37-s + 6·41-s − 4·43-s + 49-s − 6·53-s + 12·59-s − 10·61-s − 4·65-s − 4·67-s + 12·71-s − 4·73-s + 8·79-s + 12·83-s + 6·85-s + 6·89-s − 4·91-s + 2·95-s + 8·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.10·13-s + 1.45·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.359·31-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.824·53-s + 1.56·59-s − 1.28·61-s − 0.496·65-s − 0.488·67-s + 1.42·71-s − 0.468·73-s + 0.900·79-s + 1.31·83-s + 0.650·85-s + 0.635·89-s − 0.419·91-s + 0.205·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914635433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914635433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−9.702656770186684885314903914615, −9.021746631502673947025078500561, −7.911583881190548517133946426211, −7.37129406319337450735760234132, −6.35355546260478701947345777623, −5.33495198049368857874992991365, −4.79349389972818087173801528068, −3.42500814977311550003317733095, −2.44049010318535643770033231391, −1.09436889225174345511728084085,
1.09436889225174345511728084085, 2.44049010318535643770033231391, 3.42500814977311550003317733095, 4.79349389972818087173801528068, 5.33495198049368857874992991365, 6.35355546260478701947345777623, 7.37129406319337450735760234132, 7.911583881190548517133946426211, 9.021746631502673947025078500561, 9.702656770186684885314903914615
