| L(s) = 1 | − 3·5-s + 7-s + 3·11-s + 4·13-s + 6·17-s + 2·19-s − 3·23-s + 4·25-s + 2·31-s − 3·35-s + 9·37-s + 9·41-s − 2·43-s − 7·47-s − 6·49-s − 53-s − 9·55-s + 10·59-s − 4·61-s − 12·65-s + 17·73-s + 3·77-s + 8·79-s + 3·83-s − 18·85-s + 18·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.904·11-s + 1.10·13-s + 1.45·17-s + 0.458·19-s − 0.625·23-s + 4/5·25-s + 0.359·31-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 − 1.21·55-s + 1.30·59-s − 0.512·61-s − 1.48·65-s + 1.98·73-s + 0.341·77-s + 0.900·79-s + 0.329·83-s − 1.95·85-s + 1.90·89-s + 0.419·91-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
| \(L(1)\) |
\(\approx\) |
\(3.282910385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.282910385\) |
| \(L(\frac{3}{2})\) |
|
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.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 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.ac |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.81229499852568, −12.40693004390350, −11.84105387283099, −11.58319954918440, −11.25775134397376, −10.77805972190450, −10.12212706085596, −9.632823814680888, −9.207746747078662, −8.569663475401202, −8.073042481711839, −7.831527474700462, −7.457677629323952, −6.732883165511703, −6.132387608466929, −5.941310049069143, −4.983345676735415, −4.717127827760021, −3.865596772689409, −3.717176083951412, −3.285886183559951, −2.470361853323075, −1.653902212279633, −0.9690603716943631, −0.6430533510430642,
0.6430533510430642, 0.9690603716943631, 1.653902212279633, 2.470361853323075, 3.285886183559951, 3.717176083951412, 3.865596772689409, 4.717127827760021, 4.983345676735415, 5.941310049069143, 6.132387608466929, 6.732883165511703, 7.457677629323952, 7.831527474700462, 8.073042481711839, 8.569663475401202, 9.207746747078662, 9.632823814680888, 10.12212706085596, 10.77805972190450, 11.25775134397376, 11.58319954918440, 11.84105387283099, 12.40693004390350, 12.81229499852568
