| L(s) = 1 | + 3·5-s − 7-s − 11-s + 2·13-s − 2·17-s − 3·23-s + 4·25-s − 4·29-s − 10·31-s − 3·35-s − 7·37-s − 5·41-s − 4·43-s + 9·47-s − 6·49-s + 9·53-s − 3·55-s + 6·59-s + 6·65-s − 10·67-s + 10·71-s + 9·73-s + 77-s − 6·79-s − 9·83-s − 6·85-s − 2·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.625·23-s + 4/5·25-s − 0.742·29-s − 1.79·31-s − 0.507·35-s − 1.15·37-s − 0.780·41-s − 0.609·43-s + 1.31·47-s − 6/7·49-s + 1.23·53-s − 0.404·55-s + 0.781·59-s + 0.744·65-s − 1.22·67-s + 1.18·71-s + 1.05·73-s + 0.113·77-s − 0.675·79-s − 0.987·83-s − 0.650·85-s − 0.211·89-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
| \(L(1)\) |
\(\approx\) |
\(1.850851827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850851827\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.86935317493503, −12.74873210686165, −12.05280579453252, −11.46387949216110, −11.04921767225992, −10.45828247739465, −10.16922267510930, −9.727678564866889, −9.149696068100410, −8.839635378544816, −8.395266264763234, −7.665347336585805, −7.104652875014231, −6.754522746775306, −6.109204439788463, −5.735694511076270, −5.354278779829855, −4.840779719909611, −3.986010248216734, −3.608792531778598, −3.004400035041652, −2.126695609336077, −2.006515980805008, −1.317893207178183, −0.3555162438490880,
0.3555162438490880, 1.317893207178183, 2.006515980805008, 2.126695609336077, 3.004400035041652, 3.608792531778598, 3.986010248216734, 4.840779719909611, 5.354278779829855, 5.735694511076270, 6.109204439788463, 6.754522746775306, 7.104652875014231, 7.665347336585805, 8.395266264763234, 8.839635378544816, 9.149696068100410, 9.727678564866889, 10.16922267510930, 10.45828247739465, 11.04921767225992, 11.46387949216110, 12.05280579453252, 12.74873210686165, 12.86935317493503
