| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 7-s + 8-s + 9-s + 3·10-s − 11-s − 12-s + 3·13-s + 14-s − 3·15-s + 16-s + 17-s + 18-s − 6·19-s + 3·20-s − 21-s − 22-s − 2·23-s − 24-s + 4·25-s + 3·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s + 0.670·20-s − 0.218·21-s − 0.213·22-s − 0.417·23-s − 0.204·24-s + 4/5·25-s + 0.588·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.531126638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.531126638\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
| 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
−10.63628195994061693054483042228, −9.797357932916659112393474999279, −8.746586684079944788553347178225, −7.68109792457938912303406833193, −6.30800563104227857359835237500, −6.10447194422414474479910163496, −5.07426397265053455268026670853, −4.16594519366772777229966807424, −2.62601394831474839304198065711, −1.50297764615272148324460808222,
1.50297764615272148324460808222, 2.62601394831474839304198065711, 4.16594519366772777229966807424, 5.07426397265053455268026670853, 6.10447194422414474479910163496, 6.30800563104227857359835237500, 7.68109792457938912303406833193, 8.746586684079944788553347178225, 9.797357932916659112393474999279, 10.63628195994061693054483042228
