| L(s) = 1 | + 3·5-s − 7-s − 5·11-s − 2·13-s − 2·17-s + 6·19-s − 9·23-s + 4·25-s − 10·31-s − 3·35-s − 9·37-s + 9·41-s + 2·43-s + 7·47-s − 6·49-s + 53-s − 15·55-s + 6·61-s − 6·65-s + 2·67-s + 2·71-s + 9·73-s + 5·77-s − 8·79-s − 9·83-s − 6·85-s + 10·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 1.50·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s − 1.87·23-s + 4/5·25-s − 1.79·31-s − 0.507·35-s − 1.47·37-s + 1.40·41-s + 0.304·43-s + 1.02·47-s − 6/7·49-s + 0.137·53-s − 2.02·55-s + 0.768·61-s − 0.744·65-s + 0.244·67-s + 0.237·71-s + 1.05·73-s + 0.569·77-s − 0.900·79-s − 0.987·83-s − 0.650·85-s + 1.05·89-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−13.10532458613119, −12.80891999817953, −12.38593931308448, −11.85799189487552, −11.18904647822076, −10.76500162984240, −10.20929392737405, −9.958224419208661, −9.532884799132412, −9.094561862602759, −8.546780829354073, −7.833418988021700, −7.524762559261535, −7.033520855815479, −6.412810380833326, −5.722864313593014, −5.588418914951603, −5.180458627200927, −4.493074896467907, −3.792921439347461, −3.198210834716164, −2.593386333933128, −2.054980299943105, −1.800841299413080, −0.7109982922917031, 0,
0.7109982922917031, 1.800841299413080, 2.054980299943105, 2.593386333933128, 3.198210834716164, 3.792921439347461, 4.493074896467907, 5.180458627200927, 5.588418914951603, 5.722864313593014, 6.412810380833326, 7.033520855815479, 7.524762559261535, 7.833418988021700, 8.546780829354073, 9.094561862602759, 9.532884799132412, 9.958224419208661, 10.20929392737405, 10.76500162984240, 11.18904647822076, 11.85799189487552, 12.38593931308448, 12.80891999817953, 13.10532458613119
