| L(s) = 1 | − 2·2-s + 2·4-s − 7-s − 5·11-s + 13-s + 2·14-s − 4·16-s − 17-s − 4·19-s + 10·22-s − 5·23-s − 2·26-s − 2·28-s − 29-s + 10·31-s + 8·32-s + 2·34-s − 8·37-s + 8·38-s + 3·41-s + 5·43-s − 10·44-s + 10·46-s − 4·47-s + 49-s + 2·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.377·7-s − 1.50·11-s + 0.277·13-s + 0.534·14-s − 16-s − 0.242·17-s − 0.917·19-s + 2.13·22-s − 1.04·23-s − 0.392·26-s − 0.377·28-s − 0.185·29-s + 1.79·31-s + 1.41·32-s + 0.342·34-s − 1.31·37-s + 1.29·38-s + 0.468·41-s + 0.762·43-s − 1.50·44-s + 1.47·46-s − 0.583·47-s + 1/7·49-s + 0.277·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.47691670125352, −13.90593937338099, −13.43724928813532, −13.06415636747484, −12.40035600171118, −11.96640362598649, −11.24611093429440, −10.73648012006203, −10.37707385690411, −10.07284831636135, −9.447568690524482, −8.938207989753370, −8.370812002611518, −8.032977681861304, −7.596404014097530, −6.983343326258953, −6.330480542038779, −5.970594782522381, −5.114669141409326, −4.548889121562421, −3.985387628112124, −3.004278410691754, −2.539115702790619, −1.881684472010631, −1.158761437173728, 0, 0,
1.158761437173728, 1.881684472010631, 2.539115702790619, 3.004278410691754, 3.985387628112124, 4.548889121562421, 5.114669141409326, 5.970594782522381, 6.330480542038779, 6.983343326258953, 7.596404014097530, 8.032977681861304, 8.370812002611518, 8.938207989753370, 9.447568690524482, 10.07284831636135, 10.37707385690411, 10.73648012006203, 11.24611093429440, 11.96640362598649, 12.40035600171118, 13.06415636747484, 13.43724928813532, 13.90593937338099, 14.47691670125352
