| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 4·13-s − 15-s − 5·17-s + 6·19-s + 21-s − 4·25-s − 27-s − 10·29-s − 2·31-s − 35-s + 2·37-s + 4·39-s + 2·41-s − 13·43-s + 45-s + 47-s + 49-s + 5·51-s − 4·53-s − 6·57-s + 3·59-s + 8·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 1.21·17-s + 1.37·19-s + 0.218·21-s − 4/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s − 0.169·35-s + 0.328·37-s + 0.640·39-s + 0.312·41-s − 1.98·43-s + 0.149·45-s + 0.145·47-s + 1/7·49-s + 0.700·51-s − 0.549·53-s − 0.794·57-s + 0.390·59-s + 1.02·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2194582397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2194582397\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.20854365633895, −12.92913104447428, −12.24644938459775, −11.80235331798150, −11.36277493082973, −11.05277961249685, −10.29633946574716, −9.847566281104653, −9.575832755371657, −9.137116663499195, −8.493673224984820, −7.811026246945038, −7.233828214732090, −7.061994318213204, −6.357147070912825, −5.812200687691516, −5.413847519016336, −4.917794169795378, −4.349972367455269, −3.691427734196304, −3.147048156412044, −2.355885284016042, −1.920882773903989, −1.212735676355556, −0.1414214491357019,
0.1414214491357019, 1.212735676355556, 1.920882773903989, 2.355885284016042, 3.147048156412044, 3.691427734196304, 4.349972367455269, 4.917794169795378, 5.413847519016336, 5.812200687691516, 6.357147070912825, 7.061994318213204, 7.233828214732090, 7.811026246945038, 8.493673224984820, 9.137116663499195, 9.575832755371657, 9.847566281104653, 10.29633946574716, 11.05277961249685, 11.36277493082973, 11.80235331798150, 12.24644938459775, 12.92913104447428, 13.20854365633895
