| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s − 4·11-s + 13-s − 14-s − 16-s − 5·17-s − 4·22-s − 3·23-s + 26-s + 28-s − 4·29-s − 4·31-s + 5·32-s − 5·34-s − 4·37-s − 10·43-s + 4·44-s − 3·46-s − 47-s + 49-s − 52-s − 8·53-s + 3·56-s − 4·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s − 1.20·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s − 1.21·17-s − 0.852·22-s − 0.625·23-s + 0.196·26-s + 0.188·28-s − 0.742·29-s − 0.718·31-s + 0.883·32-s − 0.857·34-s − 0.657·37-s − 1.52·43-s + 0.603·44-s − 0.442·46-s − 0.145·47-s + 1/7·49-s − 0.138·52-s − 1.09·53-s + 0.400·56-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4232398638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4232398638\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
| 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
−15.61158459809800, −15.15838459151263, −14.43980636337335, −13.84613699947269, −13.44414939170095, −13.01125142432796, −12.52047447623369, −12.02873710140357, −11.13962389577105, −10.82398178312370, −10.05874945812526, −9.432049611353329, −9.026183340812801, −8.223966446124984, −7.874470485979701, −6.918139391607871, −6.358378900709621, −5.743641236056891, −5.068177206437183, −4.671700833750241, −3.797318184193759, −3.329785741924851, −2.530003354568784, −1.731061556467857, −0.2264984259412237,
0.2264984259412237, 1.731061556467857, 2.530003354568784, 3.329785741924851, 3.797318184193759, 4.671700833750241, 5.068177206437183, 5.743641236056891, 6.358378900709621, 6.918139391607871, 7.874470485979701, 8.223966446124984, 9.026183340812801, 9.432049611353329, 10.05874945812526, 10.82398178312370, 11.13962389577105, 12.02873710140357, 12.52047447623369, 13.01125142432796, 13.44414939170095, 13.84613699947269, 14.43980636337335, 15.15838459151263, 15.61158459809800
