| L(s) = 1 | − 2·5-s + 5·11-s + 4·13-s + 7·19-s − 23-s − 25-s + 4·31-s − 8·37-s − 5·41-s + 2·43-s − 47-s + 9·53-s − 10·55-s − 3·59-s − 3·61-s − 8·65-s + 2·67-s + 16·71-s − 4·73-s − 8·79-s − 14·83-s − 14·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.50·11-s + 1.10·13-s + 1.60·19-s − 0.208·23-s − 1/5·25-s + 0.718·31-s − 1.31·37-s − 0.780·41-s + 0.304·43-s − 0.145·47-s + 1.23·53-s − 1.34·55-s − 0.390·59-s − 0.384·61-s − 0.992·65-s + 0.244·67-s + 1.89·71-s − 0.468·73-s − 0.900·79-s − 1.53·83-s − 1.43·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.06703180923458, −13.84593562044359, −13.41101828462611, −12.54981387854372, −12.11818432814019, −11.76917275024117, −11.30300372957727, −11.00494980680850, −10.03285522418159, −9.861026205131797, −9.058085350951003, −8.658945950409026, −8.256945301837162, −7.542172621170000, −7.111941466248710, −6.591522751079893, −6.000393043649657, −5.453571014140536, −4.732653153163818, −4.101608263001396, −3.542071534485773, −3.387097805615803, −2.364190483233812, −1.351870784305193, −1.096465685271448, 0,
1.096465685271448, 1.351870784305193, 2.364190483233812, 3.387097805615803, 3.542071534485773, 4.101608263001396, 4.732653153163818, 5.453571014140536, 6.000393043649657, 6.591522751079893, 7.111941466248710, 7.542172621170000, 8.256945301837162, 8.658945950409026, 9.058085350951003, 9.861026205131797, 10.03285522418159, 11.00494980680850, 11.30300372957727, 11.76917275024117, 12.11818432814019, 12.54981387854372, 13.41101828462611, 13.84593562044359, 14.06703180923458
