| L(s) = 1 | − 3-s − 2·9-s − 3·11-s − 6·13-s + 5·17-s − 19-s + 7·23-s + 5·27-s − 2·29-s − 5·31-s + 3·33-s + 3·37-s + 6·39-s − 2·41-s − 4·43-s − 5·47-s − 5·51-s − 53-s + 57-s − 15·59-s + 5·61-s − 9·67-s − 7·69-s − 7·73-s + 79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.904·11-s − 1.66·13-s + 1.21·17-s − 0.229·19-s + 1.45·23-s + 0.962·27-s − 0.371·29-s − 0.898·31-s + 0.522·33-s + 0.493·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s − 0.729·47-s − 0.700·51-s − 0.137·53-s + 0.132·57-s − 1.95·59-s + 0.640·61-s − 1.09·67-s − 0.842·69-s − 0.819·73-s + 0.112·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.40373704300861, −13.80633271458384, −13.04579475903208, −12.80676181772934, −12.24665637037801, −11.77897245348543, −11.34636686595183, −10.70912974528585, −10.37008788823985, −9.818503043914863, −9.252361570211660, −8.784173183862590, −8.017847181236795, −7.576631083406161, −7.223709495387684, −6.490351921647446, −5.876730310396333, −5.360481954723052, −4.914135179179730, −4.603829717174844, −3.393535237676124, −3.085020063355817, −2.430722725881676, −1.679208438741799, −0.6724352550660202, 0,
0.6724352550660202, 1.679208438741799, 2.430722725881676, 3.085020063355817, 3.393535237676124, 4.603829717174844, 4.914135179179730, 5.360481954723052, 5.876730310396333, 6.490351921647446, 7.223709495387684, 7.576631083406161, 8.017847181236795, 8.784173183862590, 9.252361570211660, 9.818503043914863, 10.37008788823985, 10.70912974528585, 11.34636686595183, 11.77897245348543, 12.24665637037801, 12.80676181772934, 13.04579475903208, 13.80633271458384, 14.40373704300861
