| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 4·11-s − 14-s + 16-s − 2·17-s + 4·19-s + 20-s + 4·22-s + 8·23-s + 25-s − 28-s − 6·29-s + 8·31-s + 32-s − 2·34-s − 35-s + 2·37-s + 4·38-s + 40-s + 2·41-s − 12·43-s + 4·44-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 + 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.312·41-s − 1.82·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.599331739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.599331739\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.62815564050214, −13.32314278190866, −12.77876679002348, −12.31741619540823, −11.73438009454374, −11.29194348723269, −11.02925924367918, −10.19430720131458, −9.738758920205815, −9.329718573998970, −8.849258920230444, −8.223067532102505, −7.538903768444729, −6.980191697273924, −6.486854125020690, −6.262400910718289, −5.491332240452589, −4.897470179105863, −4.603828823046403, −3.697564683629968, −3.331565979219211, −2.786874943604142, −1.968680692984941, −1.369418673737635, −0.6726529825668873,
0.6726529825668873, 1.369418673737635, 1.968680692984941, 2.786874943604142, 3.331565979219211, 3.697564683629968, 4.603828823046403, 4.897470179105863, 5.491332240452589, 6.262400910718289, 6.486854125020690, 6.980191697273924, 7.538903768444729, 8.223067532102505, 8.849258920230444, 9.329718573998970, 9.738758920205815, 10.19430720131458, 11.02925924367918, 11.29194348723269, 11.73438009454374, 12.31741619540823, 12.77876679002348, 13.32314278190866, 13.62815564050214
