| L(s) = 1 | − 2·4-s + 5-s + 11-s + 4·16-s − 6·17-s − 2·19-s − 2·20-s − 8·23-s + 25-s − 8·29-s − 4·31-s + 7·37-s − 10·41-s + 4·43-s − 2·44-s + 11·53-s + 55-s − 59-s + 7·61-s − 8·64-s − 7·67-s + 12·68-s − 5·71-s − 3·73-s + 4·76-s − 5·79-s + 4·80-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.301·11-s + 16-s − 1.45·17-s − 0.458·19-s − 0.447·20-s − 1.66·23-s + 1/5·25-s − 1.48·29-s − 0.718·31-s + 1.15·37-s − 1.56·41-s + 0.609·43-s − 0.301·44-s + 1.51·53-s + 0.134·55-s − 0.130·59-s + 0.896·61-s − 64-s − 0.855·67-s + 1.45·68-s − 0.593·71-s − 0.351·73-s + 0.458·76-s − 0.562·79-s + 0.447·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 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
−13.19242281499780, −12.56027929935986, −12.10361008714253, −11.61681488630724, −11.08814688896445, −10.67608670905116, −10.09848762964371, −9.770548535268706, −9.299927923439839, −8.876577468554360, −8.514961317260724, −8.054683399773864, −7.462413127173812, −6.963091612658235, −6.433395679851889, −5.807897840625769, −5.634009505374593, −4.986920162436273, −4.296859373036563, −4.104983076190997, −3.654628265927598, −2.853260920108195, −2.184984865068586, −1.793416740514102, −1.084474682406587, 0, 0,
1.084474682406587, 1.793416740514102, 2.184984865068586, 2.853260920108195, 3.654628265927598, 4.104983076190997, 4.296859373036563, 4.986920162436273, 5.634009505374593, 5.807897840625769, 6.433395679851889, 6.963091612658235, 7.462413127173812, 8.054683399773864, 8.514961317260724, 8.876577468554360, 9.299927923439839, 9.770548535268706, 10.09848762964371, 10.67608670905116, 11.08814688896445, 11.61681488630724, 12.10361008714253, 12.56027929935986, 13.19242281499780
