| L(s) = 1 | + 5-s + 2·7-s + 11-s − 13-s + 2·19-s − 4·23-s + 25-s + 4·29-s − 6·31-s + 2·35-s − 6·37-s + 6·41-s − 3·49-s − 8·53-s + 55-s + 4·59-s + 2·61-s − 65-s − 10·67-s + 8·71-s − 2·73-s + 2·77-s + 4·79-s − 12·83-s + 2·89-s − 2·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.301·11-s − 0.277·13-s + 0.458·19-s − 0.834·23-s + 1/5·25-s + 0.742·29-s − 1.07·31-s + 0.338·35-s − 0.986·37-s + 0.937·41-s − 3/7·49-s − 1.09·53-s + 0.134·55-s + 0.520·59-s + 0.256·61-s − 0.124·65-s − 1.22·67-s + 0.949·71-s − 0.234·73-s + 0.227·77-s + 0.450·79-s − 1.31·83-s + 0.211·89-s − 0.209·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.99869615138917, −13.65292709822937, −12.86822967996579, −12.60449806609776, −11.90915264881318, −11.63097520309707, −10.97054059256119, −10.61964436629460, −9.952918282615084, −9.616452576303400, −8.982954077743929, −8.564891180362246, −7.947944945794691, −7.508127027485613, −6.967742323545968, −6.332616200356958, −5.845220043818607, −5.247845971811481, −4.801593373963955, −4.203663653456582, −3.557403600393628, −2.922445579517279, −2.149442644803591, −1.686305007018666, −0.9997461778864144, 0,
0.9997461778864144, 1.686305007018666, 2.149442644803591, 2.922445579517279, 3.557403600393628, 4.203663653456582, 4.801593373963955, 5.247845971811481, 5.845220043818607, 6.332616200356958, 6.967742323545968, 7.508127027485613, 7.947944945794691, 8.564891180362246, 8.982954077743929, 9.616452576303400, 9.952918282615084, 10.61964436629460, 10.97054059256119, 11.63097520309707, 11.90915264881318, 12.60449806609776, 12.86822967996579, 13.65292709822937, 13.99869615138917
