| L(s) = 1 | − 2·5-s + 2·11-s + 2·13-s − 6·17-s + 4·19-s + 23-s − 25-s − 2·29-s + 2·31-s − 10·43-s + 2·47-s + 4·53-s − 4·55-s + 8·59-s − 2·61-s − 4·65-s + 2·67-s + 8·71-s + 4·73-s − 4·79-s − 12·83-s + 12·85-s + 2·89-s − 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.371·29-s + 0.359·31-s − 1.52·43-s + 0.291·47-s + 0.549·53-s − 0.539·55-s + 1.04·59-s − 0.256·61-s − 0.496·65-s + 0.244·67-s + 0.949·71-s + 0.468·73-s − 0.450·79-s − 1.31·83-s + 1.30·85-s + 0.211·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.52903039331295, −13.08945190389823, −12.49343476337753, −11.95853345339838, −11.59109281521065, −11.17428433413528, −10.90993947235977, −10.05446728027017, −9.751559688964717, −9.071278518985599, −8.668389649349086, −8.266850359114889, −7.702817745656575, −7.153365237654599, −6.696051742007736, −6.324130447760991, −5.515802426708835, −5.123284515151496, −4.310999592411815, −4.062116298584382, −3.485663715413570, −2.919855155793686, −2.162810290584971, −1.503549305878138, −0.7629771544201513, 0,
0.7629771544201513, 1.503549305878138, 2.162810290584971, 2.919855155793686, 3.485663715413570, 4.062116298584382, 4.310999592411815, 5.123284515151496, 5.515802426708835, 6.324130447760991, 6.696051742007736, 7.153365237654599, 7.702817745656575, 8.266850359114889, 8.668389649349086, 9.071278518985599, 9.751559688964717, 10.05446728027017, 10.90993947235977, 11.17428433413528, 11.59109281521065, 11.95853345339838, 12.49343476337753, 13.08945190389823, 13.52903039331295
