| L(s) = 1 | − 7-s − 11-s + 13-s − 2·17-s − 5·19-s + 6·23-s + 2·29-s + 3·31-s − 10·37-s + 6·41-s + 3·43-s − 4·47-s − 6·49-s + 10·53-s − 4·59-s + 5·61-s + 5·67-s − 14·73-s + 77-s + 4·79-s + 6·83-s + 14·89-s − 91-s + 15·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.301·11-s + 0.277·13-s − 0.485·17-s − 1.14·19-s + 1.25·23-s + 0.371·29-s + 0.538·31-s − 1.64·37-s + 0.937·41-s + 0.457·43-s − 0.583·47-s − 6/7·49-s + 1.37·53-s − 0.520·59-s + 0.640·61-s + 0.610·67-s − 1.63·73-s + 0.113·77-s + 0.450·79-s + 0.658·83-s + 1.48·89-s − 0.104·91-s + 1.52·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−15.85493314065488, −15.51707454793651, −14.81831261655675, −14.43770481944325, −13.63366441044951, −13.17740204525697, −12.79472798326079, −12.14492770042752, −11.54307480733770, −10.84606920799866, −10.50445312731257, −9.909037506510892, −9.018726635408664, −8.825981713091600, −8.086980494199242, −7.414158232247047, −6.650044482144163, −6.400856714674538, −5.498300136194092, −4.895883184786699, −4.217464156770772, −3.490180269551738, −2.750380262356227, −2.061986824176624, −1.039653154976371, 0,
1.039653154976371, 2.061986824176624, 2.750380262356227, 3.490180269551738, 4.217464156770772, 4.895883184786699, 5.498300136194092, 6.400856714674538, 6.650044482144163, 7.414158232247047, 8.086980494199242, 8.825981713091600, 9.018726635408664, 9.909037506510892, 10.50445312731257, 10.84606920799866, 11.54307480733770, 12.14492770042752, 12.79472798326079, 13.17740204525697, 13.63366441044951, 14.43770481944325, 14.81831261655675, 15.51707454793651, 15.85493314065488
