| L(s) = 1 | − 3·3-s − 2·4-s + 2·5-s + 7-s + 6·9-s + 4·11-s + 6·12-s − 6·13-s − 6·15-s + 4·16-s − 7·17-s + 2·19-s − 4·20-s − 3·21-s − 7·23-s − 25-s − 9·27-s − 2·28-s − 8·29-s + 6·31-s − 12·33-s + 2·35-s − 12·36-s − 2·37-s + 18·39-s + 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s + 0.894·5-s + 0.377·7-s + 2·9-s + 1.20·11-s + 1.73·12-s − 1.66·13-s − 1.54·15-s + 16-s − 1.69·17-s + 0.458·19-s − 0.894·20-s − 0.654·21-s − 1.45·23-s − 1/5·25-s − 1.73·27-s − 0.377·28-s − 1.48·29-s + 1.07·31-s − 2.08·33-s + 0.338·35-s − 2·36-s − 0.328·37-s + 2.88·39-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 467 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
| 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
−10.56008739913459582142146844629, −9.696654753499776665780621827887, −9.238545729441004936060578659156, −7.67757313860881711666039536149, −6.52135648790445694751691086097, −5.81286116106470491906939419796, −4.85540010940614505787874939711, −4.25521155483662332316320381526, −1.77024146308666878291968722250, 0,
1.77024146308666878291968722250, 4.25521155483662332316320381526, 4.85540010940614505787874939711, 5.81286116106470491906939419796, 6.52135648790445694751691086097, 7.67757313860881711666039536149, 9.238545729441004936060578659156, 9.696654753499776665780621827887, 10.56008739913459582142146844629
