| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 3·9-s − 10-s + 6·13-s + 16-s − 17-s − 3·18-s − 2·19-s − 20-s − 2·23-s + 25-s + 6·26-s − 2·29-s + 4·31-s + 32-s − 34-s − 3·36-s − 2·37-s − 2·38-s − 40-s − 12·41-s − 8·43-s + 3·45-s − 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 9-s − 0.316·10-s + 1.66·13-s + 1/4·16-s − 0.242·17-s − 0.707·18-s − 0.458·19-s − 0.223·20-s − 0.417·23-s + 1/5·25-s + 1.17·26-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s − 1/2·36-s − 0.328·37-s − 0.324·38-s − 0.158·40-s − 1.87·41-s − 1.21·43-s + 0.447·45-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
| 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
−15.89491891263591, −15.20259437300718, −14.89628192910194, −14.22376797666076, −13.59567842188993, −13.36533897841929, −12.71475689195459, −11.88406286023888, −11.60846136570883, −11.16763062433318, −10.47377762159308, −10.02624207790389, −8.940181831496999, −8.510329943328897, −8.186339307255608, −7.320190224402118, −6.474208501373484, −6.298284115019753, −5.413043769019309, −4.989578367367883, −3.916266249003012, −3.711236572604576, −2.914751283410063, −2.110665324412919, −1.183059895390515, 0,
1.183059895390515, 2.110665324412919, 2.914751283410063, 3.711236572604576, 3.916266249003012, 4.989578367367883, 5.413043769019309, 6.298284115019753, 6.474208501373484, 7.320190224402118, 8.186339307255608, 8.510329943328897, 8.940181831496999, 10.02624207790389, 10.47377762159308, 11.16763062433318, 11.60846136570883, 11.88406286023888, 12.71475689195459, 13.36533897841929, 13.59567842188993, 14.22376797666076, 14.89628192910194, 15.20259437300718, 15.89491891263591
