| L(s) = 1 | − 5-s − 3·7-s − 3·11-s + 2·17-s + 6·19-s + 3·23-s − 4·25-s + 4·29-s + 6·31-s + 3·35-s − 7·37-s − 5·41-s + 6·43-s − 9·47-s + 2·49-s − 11·53-s + 3·55-s − 6·59-s − 8·61-s + 12·71-s − 15·73-s + 9·77-s − 12·79-s − 3·83-s − 2·85-s − 2·89-s − 6·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.904·11-s + 0.485·17-s + 1.37·19-s + 0.625·23-s − 4/5·25-s + 0.742·29-s + 1.07·31-s + 0.507·35-s − 1.15·37-s − 0.780·41-s + 0.914·43-s − 1.31·47-s + 2/7·49-s − 1.51·53-s + 0.404·55-s − 0.781·59-s − 1.02·61-s + 1.42·71-s − 1.75·73-s + 1.02·77-s − 1.35·79-s − 0.329·83-s − 0.216·85-s − 0.211·89-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4530295547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4530295547\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.04289802349984, −12.28905510118579, −12.17512681409206, −11.62987901886812, −11.07162450185557, −10.55343194157840, −10.08978196439595, −9.663309175271190, −9.380434672970326, −8.672014273210171, −8.095875798109498, −7.799523753495637, −7.263015463007617, −6.709046657504822, −6.341215042530045, −5.631035819833881, −5.279983101009170, −4.682316449781438, −4.130134036450559, −3.336383034082876, −3.053654813436527, −2.758548491308953, −1.701081533097998, −1.120994190671749, −0.1988455732244464,
0.1988455732244464, 1.120994190671749, 1.701081533097998, 2.758548491308953, 3.053654813436527, 3.336383034082876, 4.130134036450559, 4.682316449781438, 5.279983101009170, 5.631035819833881, 6.341215042530045, 6.709046657504822, 7.263015463007617, 7.799523753495637, 8.095875798109498, 8.672014273210171, 9.380434672970326, 9.663309175271190, 10.08978196439595, 10.55343194157840, 11.07162450185557, 11.62987901886812, 12.17512681409206, 12.28905510118579, 13.04289802349984
