| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 6·11-s − 13-s + 14-s + 16-s − 4·17-s + 8·19-s − 20-s + 6·22-s + 8·23-s + 25-s − 26-s + 28-s + 2·29-s − 6·31-s + 32-s − 4·34-s − 35-s + 6·37-s + 8·38-s − 40-s − 6·43-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 + 1.80·11-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 + 1.27·22-s + 1.66·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 0.371·29-s − 1.07·31-s + 0.176·32-s − 0.685·34-s − 0.169·35-s + 0.986·37-s + 1.29·38-s − 0.158·40-s − 0.914·43-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
| \(L(1)\) |
\(\approx\) |
\(3.930285617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.930285617\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−7.46014967499755461040700177781, −7.16126311070972583713256231939, −6.46316360681846536975201599552, −5.68495964347406674439645451175, −4.82945671505728617326534232155, −4.38977965724617456739179028873, −3.50843977366428260763909240096, −2.95726555220118974098632353190, −1.73290363067052476205715027584, −0.947056838645445870177592023373,
0.947056838645445870177592023373, 1.73290363067052476205715027584, 2.95726555220118974098632353190, 3.50843977366428260763909240096, 4.38977965724617456739179028873, 4.82945671505728617326534232155, 5.68495964347406674439645451175, 6.46316360681846536975201599552, 7.16126311070972583713256231939, 7.46014967499755461040700177781
