| L(s) = 1 | + 2·5-s − 2·11-s + 13-s − 6·17-s + 4·19-s − 2·23-s − 25-s − 4·31-s + 2·37-s + 2·41-s + 4·43-s + 12·53-s − 4·55-s − 6·61-s + 2·65-s + 12·67-s − 2·71-s + 2·73-s + 4·83-s − 12·85-s + 2·89-s + 8·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 0.417·23-s − 1/5·25-s − 0.718·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.64·53-s − 0.539·55-s − 0.768·61-s + 0.248·65-s + 1.46·67-s − 0.237·71-s + 0.234·73-s + 0.439·83-s − 1.30·85-s + 0.211·89-s + 0.820·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.92424678105164, −13.61929151873349, −13.15735752634023, −12.82109796998035, −12.11045477535186, −11.60789577208051, −11.08440943614861, −10.58735438849120, −10.19471198425733, −9.498818925778992, −9.232545059661059, −8.702011081763350, −8.010537589453165, −7.587800741645826, −6.906830085227933, −6.457950476838531, −5.816036500190178, −5.443348846494952, −4.892439875133228, −4.140449029468237, −3.684499495819370, −2.754632101840971, −2.338785420350957, −1.757322171875628, −0.9275108780174391, 0,
0.9275108780174391, 1.757322171875628, 2.338785420350957, 2.754632101840971, 3.684499495819370, 4.140449029468237, 4.892439875133228, 5.443348846494952, 5.816036500190178, 6.457950476838531, 6.906830085227933, 7.587800741645826, 8.010537589453165, 8.702011081763350, 9.232545059661059, 9.498818925778992, 10.19471198425733, 10.58735438849120, 11.08440943614861, 11.60789577208051, 12.11045477535186, 12.82109796998035, 13.15735752634023, 13.61929151873349, 13.92424678105164
