| 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 + 4·19-s − 20-s + 2·22-s − 4·23-s + 25-s + 26-s − 28-s − 10·29-s − 6·31-s − 32-s − 4·34-s + 35-s + 6·37-s − 4·38-s + 40-s + 2·43-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 − 0.603·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.917·19-s − 0.223·20-s + 0.426·22-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 1.85·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s + 0.986·37-s − 0.648·38-s + 0.158·40-s + 0.304·43-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
| \(L(1)\) |
\(\approx\) |
\(0.8187765424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8187765424\) |
| \(L(\frac{3}{2})\) |
|
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.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 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 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 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.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 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
−7.86172877965673395221236602266, −7.30925359608274070568557733062, −6.66983534991914862336303236930, −5.56484118569708320482957705123, −5.38920652114211755539744180812, −4.05869096321962902173777700239, −3.44689266910379429397360276429, −2.61181473424212326806308749467, −1.65910979209471518797756977088, −0.49192538178864875807113600762,
0.49192538178864875807113600762, 1.65910979209471518797756977088, 2.61181473424212326806308749467, 3.44689266910379429397360276429, 4.05869096321962902173777700239, 5.38920652114211755539744180812, 5.56484118569708320482957705123, 6.66983534991914862336303236930, 7.30925359608274070568557733062, 7.86172877965673395221236602266
