| L(s) = 1 | + 2·7-s − 11-s + 7·13-s + 3·17-s − 6·19-s + 9·23-s − 3·29-s + 5·31-s + 2·37-s + 2·41-s − 3·43-s + 7·47-s − 3·49-s − 4·53-s + 4·59-s − 14·61-s + 4·67-s − 10·71-s + 4·73-s − 2·77-s − 79-s − 10·83-s − 6·89-s + 14·91-s + 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s + 1.94·13-s + 0.727·17-s − 1.37·19-s + 1.87·23-s − 0.557·29-s + 0.898·31-s + 0.328·37-s + 0.312·41-s − 0.457·43-s + 1.02·47-s − 3/7·49-s − 0.549·53-s + 0.520·59-s − 1.79·61-s + 0.488·67-s − 1.18·71-s + 0.468·73-s − 0.227·77-s − 0.112·79-s − 1.09·83-s − 0.635·89-s + 1.46·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.567640713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.567640713\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.29217299668157, −12.69632929529390, −12.43258556143064, −11.56842637563730, −11.28056058089229, −10.87023234400585, −10.53845766044288, −9.978569642926009, −9.200730789799777, −8.868025143039729, −8.401901161887626, −8.008734284706312, −7.499463756078081, −6.800637124669046, −6.406205667903890, −5.777088403794297, −5.455043216047587, −4.625240947006148, −4.379407935808851, −3.643713317849552, −3.114605688512731, −2.551283903539396, −1.641700730574965, −1.297845330022860, −0.5767264578631175,
0.5767264578631175, 1.297845330022860, 1.641700730574965, 2.551283903539396, 3.114605688512731, 3.643713317849552, 4.379407935808851, 4.625240947006148, 5.455043216047587, 5.777088403794297, 6.406205667903890, 6.800637124669046, 7.499463756078081, 8.008734284706312, 8.401901161887626, 8.868025143039729, 9.200730789799777, 9.978569642926009, 10.53845766044288, 10.87023234400585, 11.28056058089229, 11.56842637563730, 12.43258556143064, 12.69632929529390, 13.29217299668157
