| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s − 7-s − 8-s + 9-s + 10-s + 3·11-s − 2·12-s − 4·13-s + 14-s + 2·15-s + 16-s + 3·17-s − 18-s + 2·19-s − 20-s + 2·21-s − 3·22-s + 6·23-s + 2·24-s + 25-s + 4·26-s + 4·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s − 0.639·22-s + 1.25·23-s + 0.408·24-s + 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5506207550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5506207550\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−11.46744555350262087844428001993, −10.49647218507066440612709615356, −9.688417681450633387397029903213, −8.733897697438568059214347092220, −7.47402466508823037518119617944, −6.73483784989315938735561487128, −5.73299805716885755231121374807, −4.60963444019434721197685153570, −2.99576716331676506161499712038, −0.856493313433969578668782918754,
0.856493313433969578668782918754, 2.99576716331676506161499712038, 4.60963444019434721197685153570, 5.73299805716885755231121374807, 6.73483784989315938735561487128, 7.47402466508823037518119617944, 8.733897697438568059214347092220, 9.688417681450633387397029903213, 10.49647218507066440612709615356, 11.46744555350262087844428001993
