| L(s) = 1 | + 5·7-s − 6·11-s − 4·13-s + 7·17-s − 2·19-s + 2·29-s + 4·31-s + 10·37-s − 9·41-s + 3·47-s + 18·49-s + 10·53-s − 12·59-s + 2·67-s + 16·71-s − 5·73-s − 30·77-s − 15·79-s − 6·83-s − 3·89-s − 20·91-s + 13·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 1.80·11-s − 1.10·13-s + 1.69·17-s − 0.458·19-s + 0.371·29-s + 0.718·31-s + 1.64·37-s − 1.40·41-s + 0.437·47-s + 18/7·49-s + 1.37·53-s − 1.56·59-s + 0.244·67-s + 1.89·71-s − 0.585·73-s − 3.41·77-s − 1.68·79-s − 0.658·83-s − 0.317·89-s − 2.09·91-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.11331021784878, −12.38933402591119, −12.11636482302046, −11.80625839963228, −11.08391057584744, −10.78083006063948, −10.33437156069015, −9.883110369971055, −9.519415180417369, −8.578223305546923, −8.244398062903356, −7.968139890877451, −7.510695871913589, −7.211485665428224, −6.399217585265238, −5.575075357435729, −5.378106660243258, −5.005679994214074, −4.439695171860509, −4.036506545975212, −2.988504843573498, −2.683810066855324, −2.157087770701513, −1.451086910783027, −0.8755807044264437, 0,
0.8755807044264437, 1.451086910783027, 2.157087770701513, 2.683810066855324, 2.988504843573498, 4.036506545975212, 4.439695171860509, 5.005679994214074, 5.378106660243258, 5.575075357435729, 6.399217585265238, 7.211485665428224, 7.510695871913589, 7.968139890877451, 8.244398062903356, 8.578223305546923, 9.519415180417369, 9.883110369971055, 10.33437156069015, 10.78083006063948, 11.08391057584744, 11.80625839963228, 12.11636482302046, 12.38933402591119, 13.11331021784878
