| L(s) = 1 | − 3·5-s + 7-s + 3·11-s + 2·13-s − 6·17-s + 2·19-s + 3·23-s + 4·25-s − 8·29-s + 2·31-s − 3·35-s + 3·37-s − 5·41-s − 2·43-s − 5·47-s − 6·49-s − 53-s − 9·55-s − 12·59-s − 6·61-s − 6·65-s + 2·67-s + 6·71-s + 9·73-s + 3·77-s − 9·83-s + 18·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.625·23-s + 4/5·25-s − 1.48·29-s + 0.359·31-s − 0.507·35-s + 0.493·37-s − 0.780·41-s − 0.304·43-s − 0.729·47-s − 6/7·49-s − 0.137·53-s − 1.21·55-s − 1.56·59-s − 0.768·61-s − 0.744·65-s + 0.244·67-s + 0.712·71-s + 1.05·73-s + 0.341·77-s − 0.987·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.08625763979389, −12.78192145795474, −12.01769192592035, −11.73972896130336, −11.32561740442649, −10.99702556127808, −10.66635421586339, −9.759063070762242, −9.366815103182051, −8.862013274078177, −8.509733135134165, −7.894431503476747, −7.601437216583884, −7.031482729612544, −6.412355971403788, −6.277818710204771, −5.238043071037786, −4.926974427097379, −4.197695202480315, −4.018090815202960, −3.336713734772321, −2.946109961734312, −1.930726429224487, −1.539391020304152, −0.6891984623398077, 0,
0.6891984623398077, 1.539391020304152, 1.930726429224487, 2.946109961734312, 3.336713734772321, 4.018090815202960, 4.197695202480315, 4.926974427097379, 5.238043071037786, 6.277818710204771, 6.412355971403788, 7.031482729612544, 7.601437216583884, 7.894431503476747, 8.509733135134165, 8.862013274078177, 9.366815103182051, 9.759063070762242, 10.66635421586339, 10.99702556127808, 11.32561740442649, 11.73972896130336, 12.01769192592035, 12.78192145795474, 13.08625763979389
