| L(s) = 1 | − 2·5-s − 7-s + 4·11-s − 2·13-s + 17-s − 4·19-s + 4·23-s − 25-s − 2·29-s − 8·31-s + 2·35-s + 2·37-s + 2·41-s + 8·43-s + 49-s + 6·53-s − 8·55-s − 4·59-s − 2·61-s + 4·65-s − 8·67-s + 4·71-s − 6·73-s − 4·77-s + 4·83-s − 2·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.312·41-s + 1.21·43-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 0.520·59-s − 0.256·61-s + 0.496·65-s − 0.977·67-s + 0.474·71-s − 0.702·73-s − 0.455·77-s + 0.439·83-s − 0.216·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155971553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155971553\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.32985772736466, −13.66136010248960, −13.05246692525513, −12.57940169180264, −12.17496642744583, −11.72274700071018, −11.09909292115423, −10.82157871264438, −10.11655384529017, −9.401378172478337, −9.149675325354140, −8.613494887461632, −7.920076624183772, −7.357153843308881, −7.061788570199017, −6.368588979654387, −5.838689378600860, −5.204825090414601, −4.340334774188296, −4.082871210344191, −3.494898431627228, −2.821497951153909, −2.048201940139448, −1.258899629907984, −0.3780246072088773,
0.3780246072088773, 1.258899629907984, 2.048201940139448, 2.821497951153909, 3.494898431627228, 4.082871210344191, 4.340334774188296, 5.204825090414601, 5.838689378600860, 6.368588979654387, 7.061788570199017, 7.357153843308881, 7.920076624183772, 8.613494887461632, 9.149675325354140, 9.401378172478337, 10.11655384529017, 10.82157871264438, 11.09909292115423, 11.72274700071018, 12.17496642744583, 12.57940169180264, 13.05246692525513, 13.66136010248960, 14.32985772736466
