| L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s + 13-s + 3·17-s + 19-s + 3·23-s − 25-s − 2·29-s + 5·31-s − 4·35-s + 37-s − 9·41-s + 43-s + 2·47-s − 3·49-s − 6·53-s − 4·55-s − 11·59-s − 61-s + 2·65-s − 7·67-s − 14·73-s + 4·77-s + 8·79-s + 10·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s + 0.277·13-s + 0.727·17-s + 0.229·19-s + 0.625·23-s − 1/5·25-s − 0.371·29-s + 0.898·31-s − 0.676·35-s + 0.164·37-s − 1.40·41-s + 0.152·43-s + 0.291·47-s − 3/7·49-s − 0.824·53-s − 0.539·55-s − 1.43·59-s − 0.128·61-s + 0.248·65-s − 0.855·67-s − 1.63·73-s + 0.455·77-s + 0.900·79-s + 1.09·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−16.18283056206080, −15.48808903988244, −15.12678713082828, −14.32291768228848, −13.80870551238915, −13.33633292545255, −12.94984115569369, −12.28153273925486, −11.73103374709848, −10.99642262978332, −10.26496467067927, −10.06778628924856, −9.321516245338513, −8.959803081863086, −8.019421002983068, −7.609313053676732, −6.707878129985087, −6.268564677489446, −5.633637218291342, −5.098584730661523, −4.306822138568859, −3.272048798217258, −2.959047519369459, −1.979273274916489, −1.194157742470895, 0,
1.194157742470895, 1.979273274916489, 2.959047519369459, 3.272048798217258, 4.306822138568859, 5.098584730661523, 5.633637218291342, 6.268564677489446, 6.707878129985087, 7.609313053676732, 8.019421002983068, 8.959803081863086, 9.321516245338513, 10.06778628924856, 10.26496467067927, 10.99642262978332, 11.73103374709848, 12.28153273925486, 12.94984115569369, 13.33633292545255, 13.80870551238915, 14.32291768228848, 15.12678713082828, 15.48808903988244, 16.18283056206080
