| L(s) = 1 | − 7-s − 5·11-s − 13-s + 4·17-s + 8·19-s − 5·23-s + 29-s + 8·31-s + 3·37-s + 2·41-s + 43-s + 49-s + 6·53-s + 2·59-s − 8·61-s + 5·67-s − 13·71-s − 4·73-s + 5·77-s + 7·79-s − 2·83-s + 91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.50·11-s − 0.277·13-s + 0.970·17-s + 1.83·19-s − 1.04·23-s + 0.185·29-s + 1.43·31-s + 0.493·37-s + 0.312·41-s + 0.152·43-s + 1/7·49-s + 0.824·53-s + 0.260·59-s − 1.02·61-s + 0.610·67-s − 1.54·71-s − 0.468·73-s + 0.569·77-s + 0.787·79-s − 0.219·83-s + 0.104·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.150221430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150221430\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.42967541949675, −12.16902164214118, −11.86026018411622, −11.26721544015388, −10.69760046991976, −10.21915496982054, −9.894356664040884, −9.617320454705739, −9.007136792356862, −8.262865377601053, −8.010621173570957, −7.524200701111231, −7.234834663539804, −6.508803135113525, −5.954411642886134, −5.469794810310725, −5.246822651208684, −4.520518517500607, −4.067193765871478, −3.266027021227350, −2.908791230022281, −2.552717201278863, −1.734143921147086, −0.9947258068998925, −0.4402725266122535,
0.4402725266122535, 0.9947258068998925, 1.734143921147086, 2.552717201278863, 2.908791230022281, 3.266027021227350, 4.067193765871478, 4.520518517500607, 5.246822651208684, 5.469794810310725, 5.954411642886134, 6.508803135113525, 7.234834663539804, 7.524200701111231, 8.010621173570957, 8.262865377601053, 9.007136792356862, 9.617320454705739, 9.894356664040884, 10.21915496982054, 10.69760046991976, 11.26721544015388, 11.86026018411622, 12.16902164214118, 12.42967541949675
