| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 12-s + 4·13-s − 16-s − 18-s + 4·19-s − 6·23-s − 3·24-s − 4·26-s − 27-s + 29-s − 5·32-s − 36-s − 6·37-s − 4·38-s − 4·39-s − 10·43-s + 6·46-s + 48-s − 4·52-s − 2·53-s + 54-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.288·12-s + 1.10·13-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 1.25·23-s − 0.612·24-s − 0.784·26-s − 0.192·27-s + 0.185·29-s − 0.883·32-s − 1/6·36-s − 0.986·37-s − 0.648·38-s − 0.640·39-s − 1.52·43-s + 0.884·46-s + 0.144·48-s − 0.554·52-s − 0.274·53-s + 0.136·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.70160014153947, −13.55139453929427, −12.99620868414830, −12.48255552767713, −11.74495779055516, −11.55298989386276, −10.95553505901502, −10.30824684195940, −10.02182958224957, −9.676342780970209, −8.842473899862863, −8.593940701154835, −8.063448494592587, −7.562812180775547, −6.872469271528818, −6.534168236978938, −5.601424349083773, −5.481906359773369, −4.756686879399308, −4.056552658121813, −3.728983135165563, −2.982398123399844, −1.935555904259599, −1.438085707744986, −0.7590968934293165, 0,
0.7590968934293165, 1.438085707744986, 1.935555904259599, 2.982398123399844, 3.728983135165563, 4.056552658121813, 4.756686879399308, 5.481906359773369, 5.601424349083773, 6.534168236978938, 6.872469271528818, 7.562812180775547, 8.063448494592587, 8.593940701154835, 8.842473899862863, 9.676342780970209, 10.02182958224957, 10.30824684195940, 10.95553505901502, 11.55298989386276, 11.74495779055516, 12.48255552767713, 12.99620868414830, 13.55139453929427, 13.70160014153947
