| L(s) = 1 | − 2·5-s + 4·7-s − 4·13-s + 4·17-s + 6·19-s + 3·23-s − 25-s − 8·29-s + 5·31-s − 8·35-s + 7·37-s + 2·41-s + 4·43-s − 11·47-s + 9·49-s − 53-s + 3·59-s + 8·61-s + 8·65-s + 12·67-s − 13·71-s − 14·79-s − 4·83-s − 8·85-s + 11·89-s − 16·91-s − 12·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.10·13-s + 0.970·17-s + 1.37·19-s + 0.625·23-s − 1/5·25-s − 1.48·29-s + 0.898·31-s − 1.35·35-s + 1.15·37-s + 0.312·41-s + 0.609·43-s − 1.60·47-s + 9/7·49-s − 0.137·53-s + 0.390·59-s + 1.02·61-s + 0.992·65-s + 1.46·67-s − 1.54·71-s − 1.57·79-s − 0.439·83-s − 0.867·85-s + 1.16·89-s − 1.67·91-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.51537895046599, −13.05968905131508, −12.44686221118460, −11.95636229073314, −11.62830264610087, −11.26134897571049, −10.94776686236362, −10.08841678798907, −9.688895954225967, −9.368355943860704, −8.435561961896327, −8.203066225975498, −7.640939785902469, −7.391448053912845, −7.009207427039877, −6.002791809855345, −5.507553626561595, −5.028368853576237, −4.634214069632618, −4.038647417531017, −3.470155848908662, −2.818212886559707, −2.225274489707875, −1.392184620709668, −0.9483697855929039, 0,
0.9483697855929039, 1.392184620709668, 2.225274489707875, 2.818212886559707, 3.470155848908662, 4.038647417531017, 4.634214069632618, 5.028368853576237, 5.507553626561595, 6.002791809855345, 7.009207427039877, 7.391448053912845, 7.640939785902469, 8.203066225975498, 8.435561961896327, 9.368355943860704, 9.688895954225967, 10.08841678798907, 10.94776686236362, 11.26134897571049, 11.62830264610087, 11.95636229073314, 12.44686221118460, 13.05968905131508, 13.51537895046599
