| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s + 4·13-s + 2·15-s − 17-s + 2·19-s + 2·21-s − 2·23-s − 25-s − 27-s + 6·29-s − 8·31-s + 4·35-s − 2·37-s − 4·39-s + 2·41-s + 2·43-s − 2·45-s − 4·47-s − 3·49-s + 51-s − 12·53-s − 2·57-s − 10·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 0.242·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.640·39-s + 0.312·41-s + 0.304·43-s − 0.298·45-s − 0.583·47-s − 3/7·49-s + 0.140·51-s − 1.64·53-s − 0.264·57-s − 1.30·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.56946874996271, −12.22004683140717, −11.76617713307362, −11.25446174254624, −11.02102345282148, −10.50488855999606, −10.02533674025793, −9.527953921122925, −9.051918316556333, −8.610762133633881, −8.051954870128139, −7.603229907185894, −7.232033135872870, −6.562089511057831, −6.237481212800237, −5.868636353554856, −5.211233630562371, −4.707432920915127, −4.067307578440027, −3.827948902322906, −3.114698080512724, −2.884508721506372, −1.758595980125764, −1.423956990944254, −0.5262333858311785, 0,
0.5262333858311785, 1.423956990944254, 1.758595980125764, 2.884508721506372, 3.114698080512724, 3.827948902322906, 4.067307578440027, 4.707432920915127, 5.211233630562371, 5.868636353554856, 6.237481212800237, 6.562089511057831, 7.232033135872870, 7.603229907185894, 8.051954870128139, 8.610762133633881, 9.051918316556333, 9.527953921122925, 10.02533674025793, 10.50488855999606, 11.02102345282148, 11.25446174254624, 11.76617713307362, 12.22004683140717, 12.56946874996271
