| L(s) = 1 | + 4·7-s + 11-s + 6·13-s + 2·17-s − 4·19-s − 6·23-s + 2·29-s − 8·31-s + 8·37-s − 6·41-s + 12·43-s − 10·47-s + 9·49-s − 4·59-s − 10·61-s + 2·67-s − 8·71-s − 2·73-s + 4·77-s − 4·79-s + 4·83-s + 14·89-s + 24·91-s + 4·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.301·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.25·23-s + 0.371·29-s − 1.43·31-s + 1.31·37-s − 0.937·41-s + 1.82·43-s − 1.45·47-s + 9/7·49-s − 0.520·59-s − 1.28·61-s + 0.244·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.450·79-s + 0.439·83-s + 1.48·89-s + 2.51·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.262576083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.262576083\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−14.75399690047446, −14.30263459735231, −13.88559629405687, −13.25521933128744, −12.78993988801142, −12.02650593407746, −11.67907698085934, −10.97805171475588, −10.83661568639745, −10.20860300351674, −9.344747013605740, −8.920282923462631, −8.224960630068043, −8.016458785688037, −7.416846157807032, −6.555336449893260, −5.984721601612912, −5.624914889765915, −4.732878629877993, −4.245340113096026, −3.739568267734295, −2.929507350263144, −1.807583563809322, −1.660916410543522, −0.6765698902685878,
0.6765698902685878, 1.660916410543522, 1.807583563809322, 2.929507350263144, 3.739568267734295, 4.245340113096026, 4.732878629877993, 5.624914889765915, 5.984721601612912, 6.555336449893260, 7.416846157807032, 8.016458785688037, 8.224960630068043, 8.920282923462631, 9.344747013605740, 10.20860300351674, 10.83661568639745, 10.97805171475588, 11.67907698085934, 12.02650593407746, 12.78993988801142, 13.25521933128744, 13.88559629405687, 14.30263459735231, 14.75399690047446
