| L(s) = 1 | − 3-s − 4·5-s − 4·7-s + 9-s − 2·11-s + 13-s + 4·15-s − 6·17-s + 4·21-s + 4·23-s + 11·25-s − 27-s + 6·29-s − 8·31-s + 2·33-s + 16·35-s + 10·37-s − 39-s + 4·41-s − 4·43-s − 4·45-s − 6·47-s + 9·49-s + 6·51-s − 6·53-s + 8·55-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 1.03·15-s − 1.45·17-s + 0.872·21-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s + 2.70·35-s + 1.64·37-s − 0.160·39-s + 0.624·41-s − 0.609·43-s − 0.596·45-s − 0.875·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 1.07·55-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.61965595104171, −13.15447384539823, −12.85451498344814, −12.49626437165865, −11.93076381278035, −11.33715101583635, −11.07035755420482, −10.67137736518836, −10.03237760147867, −9.410813592343137, −8.978548266049624, −8.387772317182426, −7.900992891045372, −7.286713201631779, −6.851372540541041, −6.518819829607956, −5.887791793296656, −5.192688504082662, −4.454864708344456, −4.253457065829525, −3.534074146837391, −2.988146156231579, −2.577557683400409, −1.356720496586053, −0.4647173418207874, 0,
0.4647173418207874, 1.356720496586053, 2.577557683400409, 2.988146156231579, 3.534074146837391, 4.253457065829525, 4.454864708344456, 5.192688504082662, 5.887791793296656, 6.518819829607956, 6.851372540541041, 7.286713201631779, 7.900992891045372, 8.387772317182426, 8.978548266049624, 9.410813592343137, 10.03237760147867, 10.67137736518836, 11.07035755420482, 11.33715101583635, 11.93076381278035, 12.49626437165865, 12.85451498344814, 13.15447384539823, 13.61965595104171
