| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s − 3·13-s + 15-s + 2·17-s − 5·19-s − 21-s + 4·23-s − 4·25-s − 27-s + 3·29-s − 6·31-s + 33-s − 35-s − 3·37-s + 3·39-s − 8·43-s − 45-s − 9·47-s + 49-s − 2·51-s − 4·53-s + 55-s + 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.258·15-s + 0.485·17-s − 1.14·19-s − 0.218·21-s + 0.834·23-s − 4/5·25-s − 0.192·27-s + 0.557·29-s − 1.07·31-s + 0.174·33-s − 0.169·35-s − 0.493·37-s + 0.480·39-s − 1.21·43-s − 0.149·45-s − 1.31·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s + 0.134·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−9.813316721236435331235212854281, −8.768508233439270704413769032615, −7.87516801240662189343842095601, −7.16081950157523724824352413459, −6.19273549798060271931246264625, −5.15116053426488447776293345555, −4.47152336302433330135608006264, −3.24071658344038970888982833557, −1.79936408850342859588678020613, 0,
1.79936408850342859588678020613, 3.24071658344038970888982833557, 4.47152336302433330135608006264, 5.15116053426488447776293345555, 6.19273549798060271931246264625, 7.16081950157523724824352413459, 7.87516801240662189343842095601, 8.768508233439270704413769032615, 9.813316721236435331235212854281
