| L(s) = 1 | − 3-s − 5-s − 2·9-s − 4·11-s − 7·13-s + 15-s + 4·17-s − 6·19-s + 6·23-s − 4·25-s + 5·27-s − 4·29-s − 8·31-s + 4·33-s − 10·37-s + 7·39-s + 8·41-s + 8·43-s + 2·45-s + 3·47-s − 4·51-s − 2·53-s + 4·55-s + 6·57-s + 59-s + 7·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.20·11-s − 1.94·13-s + 0.258·15-s + 0.970·17-s − 1.37·19-s + 1.25·23-s − 4/5·25-s + 0.962·27-s − 0.742·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s + 1.12·39-s + 1.24·41-s + 1.21·43-s + 0.298·45-s + 0.437·47-s − 0.560·51-s − 0.274·53-s + 0.539·55-s + 0.794·57-s + 0.130·59-s + 0.868·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.06825466265376, −12.72306876439337, −12.41670156308918, −12.02952500172818, −11.44176877284235, −10.99355104842901, −10.57091788402985, −10.29792033791776, −9.628733061315892, −9.068792105755256, −8.784149918623917, −7.940392071757580, −7.652603342953529, −7.325148154797884, −6.784356143993790, −6.046697236770885, −5.552774351887334, −5.218246108910266, −4.830793636847167, −4.175836608212876, −3.524075265234805, −2.956525284820833, −2.339454800259373, −2.021976163947157, −0.8840738392809809, 0, 0,
0.8840738392809809, 2.021976163947157, 2.339454800259373, 2.956525284820833, 3.524075265234805, 4.175836608212876, 4.830793636847167, 5.218246108910266, 5.552774351887334, 6.046697236770885, 6.784356143993790, 7.325148154797884, 7.652603342953529, 7.940392071757580, 8.784149918623917, 9.068792105755256, 9.628733061315892, 10.29792033791776, 10.57091788402985, 10.99355104842901, 11.44176877284235, 12.02952500172818, 12.41670156308918, 12.72306876439337, 13.06825466265376
