| L(s) = 1 | + 3-s − 3·5-s − 4·7-s − 2·9-s + 2·13-s − 3·15-s + 4·17-s − 6·19-s − 4·21-s + 3·23-s + 4·25-s − 5·27-s + 29-s + 9·31-s + 12·35-s − 7·37-s + 2·39-s + 8·41-s − 8·43-s + 6·45-s − 8·47-s + 9·49-s + 4·51-s − 6·53-s − 6·57-s − 7·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 1.51·7-s − 2/3·9-s + 0.554·13-s − 0.774·15-s + 0.970·17-s − 1.37·19-s − 0.872·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s + 0.185·29-s + 1.61·31-s + 2.02·35-s − 1.15·37-s + 0.320·39-s + 1.24·41-s − 1.21·43-s + 0.894·45-s − 1.16·47-s + 9/7·49-s + 0.560·51-s − 0.824·53-s − 0.794·57-s − 0.911·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−14.81030200939035, −14.13571192445480, −13.58773367443705, −13.13936715308033, −12.55996362305714, −12.16198542408608, −11.69667547307785, −11.05085961306472, −10.58669432069745, −9.973312405532090, −9.423453352982670, −8.857321028988557, −8.327068421573641, −8.044581653480860, −7.384200605039726, −6.701019983273954, −6.287803144898635, −5.763323961513529, −4.807370230637878, −4.265455153892470, −3.515699262198488, −3.206360406796186, −2.837040642659042, −1.795115835850592, −0.6788206564043664, 0,
0.6788206564043664, 1.795115835850592, 2.837040642659042, 3.206360406796186, 3.515699262198488, 4.265455153892470, 4.807370230637878, 5.763323961513529, 6.287803144898635, 6.701019983273954, 7.384200605039726, 8.044581653480860, 8.327068421573641, 8.857321028988557, 9.423453352982670, 9.973312405532090, 10.58669432069745, 11.05085961306472, 11.69667547307785, 12.16198542408608, 12.55996362305714, 13.13936715308033, 13.58773367443705, 14.13571192445480, 14.81030200939035
