| L(s) = 1 | + 5-s − 2·13-s + 2·17-s + 4·19-s + 25-s + 2·29-s + 2·37-s + 2·41-s + 12·43-s + 8·47-s − 7·49-s + 6·53-s + 12·59-s + 6·61-s − 2·65-s + 4·67-s + 6·73-s − 16·79-s + 4·83-s + 2·85-s − 10·89-s + 4·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.328·37-s + 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s + 0.824·53-s + 1.56·59-s + 0.768·61-s − 0.248·65-s + 0.488·67-s + 0.702·73-s − 1.80·79-s + 0.439·83-s + 0.216·85-s − 1.05·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.92578195461612, −12.29559881876690, −11.89474820804339, −11.48309527991384, −10.92230852379514, −10.52565258811736, −9.870995571811662, −9.739943452439710, −9.242139298233560, −8.636301955301222, −8.283783809287819, −7.571378986927915, −7.307943477300903, −6.862831385390535, −6.122913957503264, −5.839266210516447, −5.176311572695323, −5.006227482558708, −4.093562566584346, −3.867933539652256, −3.071927468283494, −2.496718788402939, −2.259485459365806, −1.177636828671172, −0.9963531473400068, 0,
0.9963531473400068, 1.177636828671172, 2.259485459365806, 2.496718788402939, 3.071927468283494, 3.867933539652256, 4.093562566584346, 5.006227482558708, 5.176311572695323, 5.839266210516447, 6.122913957503264, 6.862831385390535, 7.307943477300903, 7.571378986927915, 8.283783809287819, 8.636301955301222, 9.242139298233560, 9.739943452439710, 9.870995571811662, 10.52565258811736, 10.92230852379514, 11.48309527991384, 11.89474820804339, 12.29559881876690, 12.92578195461612
