| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 13-s − 14-s + 16-s − 2·17-s + 4·19-s + 20-s + 8·23-s + 25-s + 26-s + 28-s − 10·29-s + 4·31-s − 32-s + 2·34-s + 35-s + 10·37-s − 4·38-s − 40-s − 6·41-s − 8·46-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 1.85·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 1.17·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706378846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706378846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.71051564543209933535935466982, −7.29374187461655957576075016952, −6.58894372928010091161412620387, −5.75411905218635527276875673041, −5.15995623323811083702438577658, −4.34508854765871894206559153521, −3.27242650566967892437314634364, −2.53034116885894039152679399556, −1.64971253501805017577982325693, −0.74083214778129269410623102876,
0.74083214778129269410623102876, 1.64971253501805017577982325693, 2.53034116885894039152679399556, 3.27242650566967892437314634364, 4.34508854765871894206559153521, 5.15995623323811083702438577658, 5.75411905218635527276875673041, 6.58894372928010091161412620387, 7.29374187461655957576075016952, 7.71051564543209933535935466982
