| L(s) = 1 | − 2·4-s − 2·5-s − 7-s + 11-s − 13-s + 4·16-s − 2·17-s + 4·20-s + 4·23-s − 25-s + 2·28-s − 9·29-s + 2·31-s + 2·35-s + 9·37-s + 7·41-s − 5·43-s − 2·44-s + 10·47-s + 49-s + 2·52-s − 9·53-s − 2·55-s − 4·59-s − 2·61-s − 8·64-s + 2·65-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s − 0.377·7-s + 0.301·11-s − 0.277·13-s + 16-s − 0.485·17-s + 0.894·20-s + 0.834·23-s − 1/5·25-s + 0.377·28-s − 1.67·29-s + 0.359·31-s + 0.338·35-s + 1.47·37-s + 1.09·41-s − 0.762·43-s − 0.301·44-s + 1.45·47-s + 1/7·49-s + 0.277·52-s − 1.23·53-s − 0.269·55-s − 0.520·59-s − 0.256·61-s − 64-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.13153652817093, −12.76745937897756, −12.29782849331629, −11.80908238529860, −11.34469461055961, −10.92360987235764, −10.44369278664052, −9.757072132507051, −9.456571020617996, −9.013814038985537, −8.673882457245285, −8.002514952011950, −7.529381360709509, −7.385249656115032, −6.575362719789832, −6.040753873487696, −5.595971445218213, −4.999839585792486, −4.379487750230815, −4.121411479096243, −3.694765454999392, −2.955159632115530, −2.599377740646125, −1.552496994244083, −1.033442004649813, 0, 0,
1.033442004649813, 1.552496994244083, 2.599377740646125, 2.955159632115530, 3.694765454999392, 4.121411479096243, 4.379487750230815, 4.999839585792486, 5.595971445218213, 6.040753873487696, 6.575362719789832, 7.385249656115032, 7.529381360709509, 8.002514952011950, 8.673882457245285, 9.013814038985537, 9.456571020617996, 9.757072132507051, 10.44369278664052, 10.92360987235764, 11.34469461055961, 11.80908238529860, 12.29782849331629, 12.76745937897756, 13.13153652817093
