| L(s) = 1 | − 5-s + 2·7-s − 13-s − 2·17-s + 8·19-s − 8·23-s + 25-s − 4·31-s − 2·35-s + 2·37-s + 4·41-s − 2·43-s − 3·49-s − 6·53-s − 4·59-s + 4·61-s + 65-s + 4·67-s − 8·71-s − 2·73-s + 8·79-s − 14·83-s + 2·85-s + 10·89-s − 2·91-s − 8·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.277·13-s − 0.485·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.624·41-s − 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.520·59-s + 0.512·61-s + 0.124·65-s + 0.488·67-s − 0.949·71-s − 0.234·73-s + 0.900·79-s − 1.53·83-s + 0.216·85-s + 1.05·89-s − 0.209·91-s − 0.820·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.85546172538257, −12.51596667639370, −11.84672220311053, −11.64029701563295, −11.31155380851590, −10.70026676374289, −10.23773405326371, −9.721768232635935, −9.280102437878517, −8.859114854190901, −8.120724363276401, −7.831444885303089, −7.568429964383784, −6.926603553426551, −6.412571263808626, −5.769608729423242, −5.357972798048876, −4.841611122734662, −4.318390932371862, −3.854816090814228, −3.241485581240695, −2.716093667613140, −1.956500066186182, −1.533481880620136, −0.7543474445674610, 0,
0.7543474445674610, 1.533481880620136, 1.956500066186182, 2.716093667613140, 3.241485581240695, 3.854816090814228, 4.318390932371862, 4.841611122734662, 5.357972798048876, 5.769608729423242, 6.412571263808626, 6.926603553426551, 7.568429964383784, 7.831444885303089, 8.120724363276401, 8.859114854190901, 9.280102437878517, 9.721768232635935, 10.23773405326371, 10.70026676374289, 11.31155380851590, 11.64029701563295, 11.84672220311053, 12.51596667639370, 12.85546172538257
