| L(s) = 1 | − 2·5-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s − 25-s − 6·29-s + 8·31-s − 6·37-s + 6·41-s − 4·43-s − 7·49-s − 2·53-s − 8·55-s − 4·59-s + 2·61-s + 4·65-s + 4·67-s − 8·71-s + 10·73-s + 8·79-s − 4·83-s − 4·85-s − 6·89-s − 8·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 49-s − 0.274·53-s − 1.07·55-s − 0.520·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.900·79-s − 0.439·83-s − 0.433·85-s − 0.635·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−15.09057331342915, −14.62914657807155, −14.05768292978718, −13.72453912879758, −12.91908740709758, −12.31715446233378, −11.96758469690085, −11.50371131555606, −11.11771689829495, −10.31538347458514, −9.645580648027357, −9.432479272376017, −8.634972723171745, −8.080164510220657, −7.565960608643644, −7.085518347512728, −6.457815794338902, −5.842276700168263, −5.081161269800029, −4.554971623344801, −3.779282832081591, −3.472152063580733, −2.657092392379667, −1.697908415098429, −0.9808311582167849, 0,
0.9808311582167849, 1.697908415098429, 2.657092392379667, 3.472152063580733, 3.779282832081591, 4.554971623344801, 5.081161269800029, 5.842276700168263, 6.457815794338902, 7.085518347512728, 7.565960608643644, 8.080164510220657, 8.634972723171745, 9.432479272376017, 9.645580648027357, 10.31538347458514, 11.11771689829495, 11.50371131555606, 11.96758469690085, 12.31715446233378, 12.91908740709758, 13.72453912879758, 14.05768292978718, 14.62914657807155, 15.09057331342915
