| L(s) = 1 | − 2-s − 3·3-s + 4-s + 2·5-s + 3·6-s − 7-s − 8-s + 6·9-s − 2·10-s + 2·11-s − 3·12-s + 3·13-s + 14-s − 6·15-s + 16-s − 6·18-s + 8·19-s + 2·20-s + 3·21-s − 2·22-s − 4·23-s + 3·24-s − 25-s − 3·26-s − 9·27-s − 28-s − 10·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.894·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 0.632·10-s + 0.603·11-s − 0.866·12-s + 0.832·13-s + 0.267·14-s − 1.54·15-s + 1/4·16-s − 1.41·18-s + 1.83·19-s + 0.447·20-s + 0.654·21-s − 0.426·22-s − 0.834·23-s + 0.612·24-s − 1/5·25-s − 0.588·26-s − 1.73·27-s − 0.188·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−16.06438164984531, −15.50424691362195, −14.75523988629122, −13.94010326156533, −13.51035363197366, −12.88936171999698, −12.28945380626753, −11.66756807555535, −11.45494370777399, −10.75538083975835, −10.27028410222538, −9.770077661128787, −9.254278244347153, −8.798184972627312, −7.571157078393439, −7.301654240940515, −6.621510675274557, −5.903191641246684, −5.671822744279666, −5.292987160313405, −4.036264190208722, −3.654188229502138, −2.383341457374383, −1.522054074190632, −1.005294514377508, 0,
1.005294514377508, 1.522054074190632, 2.383341457374383, 3.654188229502138, 4.036264190208722, 5.292987160313405, 5.671822744279666, 5.903191641246684, 6.621510675274557, 7.301654240940515, 7.571157078393439, 8.798184972627312, 9.254278244347153, 9.770077661128787, 10.27028410222538, 10.75538083975835, 11.45494370777399, 11.66756807555535, 12.28945380626753, 12.88936171999698, 13.51035363197366, 13.94010326156533, 14.75523988629122, 15.50424691362195, 16.06438164984531
