| L(s) = 1 | − 2·5-s + 2·11-s − 4·17-s + 4·19-s − 6·23-s − 25-s − 6·29-s − 31-s − 10·37-s + 4·41-s + 43-s + 10·47-s − 8·53-s − 4·55-s + 2·59-s + 5·61-s + 7·67-s + 10·71-s − 7·73-s + 17·79-s − 12·83-s + 8·85-s + 16·89-s − 8·95-s + 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s − 0.970·17-s + 0.917·19-s − 1.25·23-s − 1/5·25-s − 1.11·29-s − 0.179·31-s − 1.64·37-s + 0.624·41-s + 0.152·43-s + 1.45·47-s − 1.09·53-s − 0.539·55-s + 0.260·59-s + 0.640·61-s + 0.855·67-s + 1.18·71-s − 0.819·73-s + 1.91·79-s − 1.31·83-s + 0.867·85-s + 1.69·89-s − 0.820·95-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007781421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007781421\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−12.53527389464725, −12.13229818831041, −11.83970974557067, −11.39650930819514, −10.82176322093146, −10.65897750129440, −9.729522193921788, −9.572905309521744, −8.962757905471037, −8.568643838635731, −7.944065861064497, −7.643575605225155, −7.135804527943666, −6.661964214981830, −6.157409419884871, −5.524280350265111, −5.150573819429685, −4.412846170382956, −3.893287865000102, −3.731384633530226, −3.067426780031260, −2.211459312400742, −1.883978454232703, −1.030875675265470, −0.2945796829092631,
0.2945796829092631, 1.030875675265470, 1.883978454232703, 2.211459312400742, 3.067426780031260, 3.731384633530226, 3.893287865000102, 4.412846170382956, 5.150573819429685, 5.524280350265111, 6.157409419884871, 6.661964214981830, 7.135804527943666, 7.643575605225155, 7.944065861064497, 8.568643838635731, 8.962757905471037, 9.572905309521744, 9.729522193921788, 10.65897750129440, 10.82176322093146, 11.39650930819514, 11.83970974557067, 12.13229818831041, 12.53527389464725
