| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 4·7-s + 3·8-s + 9-s − 10-s − 4·11-s − 12-s + 2·13-s + 4·14-s + 15-s − 16-s − 6·17-s − 18-s − 4·19-s − 20-s − 4·21-s + 4·22-s + 3·24-s + 25-s − 2·26-s + 27-s + 4·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26272008989189304077761027502, −9.619881912622415673569211765264, −8.894003158767714592279341847617, −8.161265561981018255582993597565, −7.02137188315382950394322884244, −6.06865597150852415394362452213, −4.69288327276767791313809475264, −3.47701333779520587015827014256, −2.17254925636267642483655615973, 0,
2.17254925636267642483655615973, 3.47701333779520587015827014256, 4.69288327276767791313809475264, 6.06865597150852415394362452213, 7.02137188315382950394322884244, 8.161265561981018255582993597565, 8.894003158767714592279341847617, 9.619881912622415673569211765264, 10.26272008989189304077761027502
