| L(s) = 1 | + 5-s + 7-s + 3·11-s − 13-s + 3·17-s − 4·19-s − 9·23-s + 25-s − 6·29-s − 2·31-s + 35-s + 37-s + 3·41-s + 2·43-s − 6·47-s − 6·49-s + 9·53-s + 3·55-s + 12·59-s − 5·61-s − 65-s − 4·67-s + 9·71-s + 14·73-s + 3·77-s + 7·79-s + 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s − 1.87·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.169·35-s + 0.164·37-s + 0.468·41-s + 0.304·43-s − 0.875·47-s − 6/7·49-s + 1.23·53-s + 0.404·55-s + 1.56·59-s − 0.640·61-s − 0.124·65-s − 0.488·67-s + 1.06·71-s + 1.63·73-s + 0.341·77-s + 0.787·79-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.01042570026719, −14.43768393328948, −14.29252108963695, −13.62408550999054, −13.03344130490881, −12.45396285004873, −12.05256763271569, −11.43452634889886, −10.98210287300761, −10.29222546969276, −9.788436479806993, −9.375014760534885, −8.749889498424843, −8.013983914799363, −7.795451604115936, −6.806375544619681, −6.514442426758775, −5.658838192499456, −5.439181598449173, −4.437500085019651, −3.984408664814212, −3.393273280704977, −2.310389061102857, −1.922779677184811, −1.107170418716238, 0,
1.107170418716238, 1.922779677184811, 2.310389061102857, 3.393273280704977, 3.984408664814212, 4.437500085019651, 5.439181598449173, 5.658838192499456, 6.514442426758775, 6.806375544619681, 7.795451604115936, 8.013983914799363, 8.749889498424843, 9.375014760534885, 9.788436479806993, 10.29222546969276, 10.98210287300761, 11.43452634889886, 12.05256763271569, 12.45396285004873, 13.03344130490881, 13.62408550999054, 14.29252108963695, 14.43768393328948, 15.01042570026719
