| L(s) = 1 | − 5-s + 3·7-s + 3·11-s + 4·13-s + 2·17-s − 6·19-s − 3·23-s − 4·25-s − 2·31-s − 3·35-s + 37-s + 3·41-s − 10·43-s − 7·47-s + 2·49-s − 3·53-s − 3·55-s − 10·59-s + 8·61-s − 4·65-s − 8·67-s + 8·71-s − 7·73-s + 9·77-s − 12·79-s + 11·83-s − 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.904·11-s + 1.10·13-s + 0.485·17-s − 1.37·19-s − 0.625·23-s − 4/5·25-s − 0.359·31-s − 0.507·35-s + 0.164·37-s + 0.468·41-s − 1.52·43-s − 1.02·47-s + 2/7·49-s − 0.412·53-s − 0.404·55-s − 1.30·59-s + 1.02·61-s − 0.496·65-s − 0.977·67-s + 0.949·71-s − 0.819·73-s + 1.02·77-s − 1.35·79-s + 1.20·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.103418188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103418188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 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 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.06934246022789, −12.27412091051369, −11.98978240380892, −11.49413136563536, −11.15811239565995, −10.75546223910885, −10.21386521403497, −9.638987889947447, −9.138841987081924, −8.495672510480457, −8.293043532988756, −7.878510283493483, −7.330513704215956, −6.630866523490684, −6.266613378361254, −5.796207718734069, −5.160989426120826, −4.501094276038322, −4.205153082265695, −3.638721919794456, −3.204298151815073, −2.241695932177073, −1.635127791132965, −1.388561146584734, −0.3966360950872783,
0.3966360950872783, 1.388561146584734, 1.635127791132965, 2.241695932177073, 3.204298151815073, 3.638721919794456, 4.205153082265695, 4.501094276038322, 5.160989426120826, 5.796207718734069, 6.266613378361254, 6.630866523490684, 7.330513704215956, 7.878510283493483, 8.293043532988756, 8.495672510480457, 9.138841987081924, 9.638987889947447, 10.21386521403497, 10.75546223910885, 11.15811239565995, 11.49413136563536, 11.98978240380892, 12.27412091051369, 13.06934246022789
