| L(s) = 1 | + 2·5-s + 2·7-s + 2·11-s − 2·13-s + 4·17-s + 23-s − 25-s + 10·29-s + 4·35-s − 4·37-s + 6·41-s − 4·43-s − 8·47-s − 3·49-s + 6·53-s + 4·55-s − 4·59-s + 8·61-s − 4·65-s − 4·67-s − 8·71-s + 6·73-s + 4·77-s + 6·79-s + 6·83-s + 8·85-s + 4·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s + 0.970·17-s + 0.208·23-s − 1/5·25-s + 1.85·29-s + 0.676·35-s − 0.657·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 1.02·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s + 0.702·73-s + 0.455·77-s + 0.675·79-s + 0.658·83-s + 0.867·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.345736392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345736392\) |
| \(L(\frac{3}{2})\) |
|
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.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 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 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−9.492318081275224432442276151661, −8.563917979768193761026400475481, −7.86439936922905516957509602013, −6.90741293334474489756798936148, −6.11540448863638237893353520402, −5.24902277752070792721028959540, −4.54757426650994595317255163463, −3.30763157128856992520002639408, −2.17956547505623602678219636554, −1.17892632554232482008097210010,
1.17892632554232482008097210010, 2.17956547505623602678219636554, 3.30763157128856992520002639408, 4.54757426650994595317255163463, 5.24902277752070792721028959540, 6.11540448863638237893353520402, 6.90741293334474489756798936148, 7.86439936922905516957509602013, 8.563917979768193761026400475481, 9.492318081275224432442276151661
