| L(s) = 1 | − 2·5-s + 4·7-s − 4·11-s − 2·13-s + 2·17-s + 23-s − 25-s + 2·29-s − 8·35-s − 10·37-s + 6·41-s − 8·43-s − 8·47-s + 9·49-s + 6·53-s + 8·55-s − 4·59-s + 14·61-s + 4·65-s − 8·67-s − 8·71-s − 6·73-s − 16·77-s − 12·79-s − 12·83-s − 4·85-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.208·23-s − 1/5·25-s + 0.371·29-s − 1.35·35-s − 1.64·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s + 1.07·55-s − 0.520·59-s + 1.79·61-s + 0.496·65-s − 0.977·67-s − 0.949·71-s − 0.702·73-s − 1.82·77-s − 1.35·79-s − 1.31·83-s − 0.433·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−8.327781090606475870382342337557, −7.50699486492617514936997772257, −7.17107479898425035723239787460, −5.78526871530178298138769664606, −5.04108692830443101591689388038, −4.56422440393451236797823788719, −3.54192240000087360659655825677, −2.53706594040917035071504886832, −1.48200439795152071424990934394, 0,
1.48200439795152071424990934394, 2.53706594040917035071504886832, 3.54192240000087360659655825677, 4.56422440393451236797823788719, 5.04108692830443101591689388038, 5.78526871530178298138769664606, 7.17107479898425035723239787460, 7.50699486492617514936997772257, 8.327781090606475870382342337557
