| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 2·11-s + 13-s + 14-s + 16-s − 4·17-s − 8·19-s + 20-s − 2·22-s + 8·23-s + 25-s + 26-s + 28-s + 6·29-s − 2·31-s + 32-s − 4·34-s + 35-s + 10·37-s − 8·38-s + 40-s − 4·41-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 − 0.603·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.83·19-s + 0.223·20-s − 0.426·22-s + 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 0.685·34-s + 0.169·35-s + 1.64·37-s − 1.29·38-s + 0.158·40-s − 0.624·41-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
| \(L(1)\) |
\(\approx\) |
\(3.665835831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.665835831\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−7.76634326984586350632372766384, −6.86057522368596310000804813497, −6.44010759220480879723926060936, −5.72059830573245017533946481346, −4.83521065442879528501507398067, −4.52552178489530673454170157045, −3.55215993859798483709183766894, −2.54221585522815788797287431399, −2.10912837995800716108955088415, −0.842857540353727687934095928810,
0.842857540353727687934095928810, 2.10912837995800716108955088415, 2.54221585522815788797287431399, 3.55215993859798483709183766894, 4.52552178489530673454170157045, 4.83521065442879528501507398067, 5.72059830573245017533946481346, 6.44010759220480879723926060936, 6.86057522368596310000804813497, 7.76634326984586350632372766384
