| L(s) = 1 | + 3·5-s − 3·9-s + 3·11-s − 5·13-s − 6·17-s + 4·19-s − 8·23-s + 4·25-s + 8·29-s + 7·31-s + 37-s − 2·41-s + 6·43-s − 9·45-s − 6·47-s − 53-s + 9·55-s − 9·59-s + 6·61-s − 15·65-s + 11·67-s − 15·71-s − 2·73-s − 4·79-s + 9·81-s + 14·83-s − 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 9-s + 0.904·11-s − 1.38·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 4/5·25-s + 1.48·29-s + 1.25·31-s + 0.164·37-s − 0.312·41-s + 0.914·43-s − 1.34·45-s − 0.875·47-s − 0.137·53-s + 1.21·55-s − 1.17·59-s + 0.768·61-s − 1.86·65-s + 1.34·67-s − 1.78·71-s − 0.234·73-s − 0.450·79-s + 81-s + 1.53·83-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−14.03415780425259, −13.42299452357764, −13.05729914415027, −12.16999287343207, −11.90525300362345, −11.67170124169493, −10.80216376181343, −10.39584894773157, −9.812094491774977, −9.454129743262771, −9.163761186705805, −8.334646576188103, −8.143333327970523, −7.277706086152804, −6.659363892096679, −6.324680040463346, −5.871005658067438, −5.303596277732780, −4.643616303807351, −4.337701873041477, −3.356279708964336, −2.700415364606021, −2.310190474354377, −1.764788344959233, −0.8895984284651878, 0,
0.8895984284651878, 1.764788344959233, 2.310190474354377, 2.700415364606021, 3.356279708964336, 4.337701873041477, 4.643616303807351, 5.303596277732780, 5.871005658067438, 6.324680040463346, 6.659363892096679, 7.277706086152804, 8.143333327970523, 8.334646576188103, 9.163761186705805, 9.454129743262771, 9.812094491774977, 10.39584894773157, 10.80216376181343, 11.67170124169493, 11.90525300362345, 12.16999287343207, 13.05729914415027, 13.42299452357764, 14.03415780425259
