| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 2·7-s − 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 4·13-s + 2·14-s − 2·15-s + 16-s + 2·17-s − 18-s + 8·19-s + 2·20-s + 2·21-s − 4·22-s + 24-s − 25-s + 4·26-s − 27-s − 2·28-s − 8·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.83·19-s + 0.447·20-s + 0.436·21-s − 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.394848526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394848526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.61030762026105, −12.89225088217233, −12.41152694530867, −11.93760793769902, −11.54469239817360, −11.20771182689237, −10.28371893838940, −9.910158163605782, −9.753501506518460, −9.279285688826088, −8.873007406669574, −7.997539057668553, −7.453771060182108, −7.101363044438416, −6.474191394690120, −6.171175982576019, −5.422590874051128, −5.248178633730254, −4.399736371313016, −3.523176502054895, −3.247894429250932, −2.362852933315944, −1.755873615825031, −1.174757849096758, −0.4446314226574568,
0.4446314226574568, 1.174757849096758, 1.755873615825031, 2.362852933315944, 3.247894429250932, 3.523176502054895, 4.399736371313016, 5.248178633730254, 5.422590874051128, 6.171175982576019, 6.474191394690120, 7.101363044438416, 7.453771060182108, 7.997539057668553, 8.873007406669574, 9.279285688826088, 9.753501506518460, 9.910158163605782, 10.28371893838940, 11.20771182689237, 11.54469239817360, 11.93760793769902, 12.41152694530867, 12.89225088217233, 13.61030762026105
