| L(s) = 1 | + 2·7-s + 11-s − 6·17-s + 4·19-s − 6·23-s − 6·29-s − 6·37-s + 10·41-s + 8·43-s + 6·47-s − 3·49-s − 12·53-s + 8·59-s + 4·61-s + 12·67-s − 10·71-s − 2·73-s + 2·77-s + 2·79-s + 12·83-s + 6·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s − 1.45·17-s + 0.917·19-s − 1.25·23-s − 1.11·29-s − 0.986·37-s + 1.56·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s − 1.64·53-s + 1.04·59-s + 0.512·61-s + 1.46·67-s − 1.18·71-s − 0.234·73-s + 0.227·77-s + 0.225·79-s + 1.31·83-s + 0.635·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.96701013038691, −15.52417731147097, −14.69104769180144, −14.38164956030395, −13.82671584083784, −13.28685966120550, −12.64601297814751, −12.05212837764954, −11.50324247888759, −10.98262083472478, −10.61370912389710, −9.630956251779411, −9.320504916783736, −8.660319126961626, −7.979857846480258, −7.528202557070149, −6.848036563667883, −6.162937813273615, −5.527312726380805, −4.892326788120019, −4.134246567904415, −3.720727423038733, −2.583601547346159, −2.012225213834002, −1.169588620108249, 0,
1.169588620108249, 2.012225213834002, 2.583601547346159, 3.720727423038733, 4.134246567904415, 4.892326788120019, 5.527312726380805, 6.162937813273615, 6.848036563667883, 7.528202557070149, 7.979857846480258, 8.660319126961626, 9.320504916783736, 9.630956251779411, 10.61370912389710, 10.98262083472478, 11.50324247888759, 12.05212837764954, 12.64601297814751, 13.28685966120550, 13.82671584083784, 14.38164956030395, 14.69104769180144, 15.52417731147097, 15.96701013038691
