| L(s) = 1 | − 3·5-s − 2·11-s + 13-s − 6·17-s − 7·19-s − 23-s + 4·25-s + 5·29-s − 9·31-s + 4·37-s + 6·41-s + 9·43-s − 7·47-s − 3·53-s + 6·55-s + 4·59-s − 2·61-s − 3·65-s − 8·67-s − 11·73-s − 9·79-s + 3·83-s + 18·85-s + 9·89-s + 21·95-s + 9·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.603·11-s + 0.277·13-s − 1.45·17-s − 1.60·19-s − 0.208·23-s + 4/5·25-s + 0.928·29-s − 1.61·31-s + 0.657·37-s + 0.937·41-s + 1.37·43-s − 1.02·47-s − 0.412·53-s + 0.809·55-s + 0.520·59-s − 0.256·61-s − 0.372·65-s − 0.977·67-s − 1.28·73-s − 1.01·79-s + 0.329·83-s + 1.95·85-s + 0.953·89-s + 2.15·95-s + 0.913·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−12.72721085449943, −12.35560497747379, −11.70518418387177, −11.38775698608119, −10.91348739209299, −10.64387045958181, −10.23012606963333, −9.425802368647512, −8.870140794084671, −8.753021356778822, −8.095931352065068, −7.700418164214915, −7.380646390309141, −6.701916499818029, −6.330936009630790, −5.855591457989212, −5.126013821470347, −4.576361787580460, −4.138827760260396, −3.965652495202856, −3.163459994651861, −2.599955191494224, −2.137622616716863, −1.404957566717876, −0.4670298368377933, 0,
0.4670298368377933, 1.404957566717876, 2.137622616716863, 2.599955191494224, 3.163459994651861, 3.965652495202856, 4.138827760260396, 4.576361787580460, 5.126013821470347, 5.855591457989212, 6.330936009630790, 6.701916499818029, 7.380646390309141, 7.700418164214915, 8.095931352065068, 8.753021356778822, 8.870140794084671, 9.425802368647512, 10.23012606963333, 10.64387045958181, 10.91348739209299, 11.38775698608119, 11.70518418387177, 12.35560497747379, 12.72721085449943
