| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 2·11-s − 12-s + 5·13-s + 16-s − 17-s + 18-s − 19-s − 2·22-s − 5·23-s − 24-s + 5·26-s − 27-s + 9·29-s − 3·31-s + 32-s + 2·33-s − 34-s + 36-s + 8·37-s − 38-s − 5·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 1.38·13-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.426·22-s − 1.04·23-s − 0.204·24-s + 0.980·26-s − 0.192·27-s + 1.67·29-s − 0.538·31-s + 0.176·32-s + 0.348·33-s − 0.171·34-s + 1/6·36-s + 1.31·37-s − 0.162·38-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.922420555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.922420555\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−13.30892882047511, −12.83555303068448, −12.69815971841908, −11.88973331852132, −11.59090498434329, −11.04983848864737, −10.66672302173732, −10.25841682940056, −9.672004615500635, −9.063152338720197, −8.346510585383165, −8.106441936713927, −7.404482144729188, −6.896769285644326, −6.228221889975694, −5.979141552569556, −5.544548985709148, −4.852189826100944, −4.169301977110431, −4.071235508647788, −3.193324534858703, −2.579556388257282, −2.022961381094035, −1.145199258196044, −0.5972602323378662,
0.5972602323378662, 1.145199258196044, 2.022961381094035, 2.579556388257282, 3.193324534858703, 4.071235508647788, 4.169301977110431, 4.852189826100944, 5.544548985709148, 5.979141552569556, 6.228221889975694, 6.896769285644326, 7.404482144729188, 8.106441936713927, 8.346510585383165, 9.063152338720197, 9.672004615500635, 10.25841682940056, 10.66672302173732, 11.04983848864737, 11.59090498434329, 11.88973331852132, 12.69815971841908, 12.83555303068448, 13.30892882047511
