| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s − 4·11-s + 2·13-s + 14-s + 16-s − 6·17-s + 2·20-s + 4·22-s − 23-s − 25-s − 2·26-s − 28-s + 2·29-s + 4·31-s − 32-s + 6·34-s − 2·35-s + 6·37-s − 2·40-s + 6·41-s + 12·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s − 1.20·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.447·20-s + 0.852·22-s − 0.208·23-s − 1/5·25-s − 0.392·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s + 0.986·37-s − 0.316·40-s + 0.937·41-s + 1.82·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.266819530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.266819530\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−8.940828619209000408651180340017, −8.035746589693562494367466631185, −7.41708567968259338700255296735, −6.31445741146935302983518168441, −6.04927408607389576770507998327, −5.01462829349377636498382770147, −3.98862924634580351907636573224, −2.63320388374996452027450715872, −2.21224587576661076823557900596, −0.74842334983695760475932304210,
0.74842334983695760475932304210, 2.21224587576661076823557900596, 2.63320388374996452027450715872, 3.98862924634580351907636573224, 5.01462829349377636498382770147, 6.04927408607389576770507998327, 6.31445741146935302983518168441, 7.41708567968259338700255296735, 8.035746589693562494367466631185, 8.940828619209000408651180340017
