| L(s) = 1 | − 5-s + 4·7-s − 3·11-s + 6·13-s + 4·17-s − 3·19-s − 23-s − 4·25-s + 4·29-s + 2·31-s − 4·35-s − 3·37-s − 6·41-s − 12·43-s + 9·49-s − 9·53-s + 3·55-s + 7·59-s + 61-s − 6·65-s + 7·67-s + 4·73-s − 12·77-s + 4·79-s − 83-s − 4·85-s − 5·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.904·11-s + 1.66·13-s + 0.970·17-s − 0.688·19-s − 0.208·23-s − 4/5·25-s + 0.742·29-s + 0.359·31-s − 0.676·35-s − 0.493·37-s − 0.937·41-s − 1.82·43-s + 9/7·49-s − 1.23·53-s + 0.404·55-s + 0.911·59-s + 0.128·61-s − 0.744·65-s + 0.855·67-s + 0.468·73-s − 1.36·77-s + 0.450·79-s − 0.109·83-s − 0.433·85-s − 0.529·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.82569896275900, −14.32464719432136, −13.80767707293004, −13.37578856334083, −12.85013057072269, −12.04862307280234, −11.75650394870226, −11.21633637958721, −10.75970285073443, −10.28893999839617, −9.760397800756041, −8.786388303944274, −8.367280496781873, −8.023706715763651, −7.747204703685402, −6.770226004962758, −6.326096159765753, −5.469583689208119, −5.156198258181670, −4.497657338509505, −3.793097976719950, −3.348762756924688, −2.404176006441524, −1.643832601072350, −1.133308764751995, 0,
1.133308764751995, 1.643832601072350, 2.404176006441524, 3.348762756924688, 3.793097976719950, 4.497657338509505, 5.156198258181670, 5.469583689208119, 6.326096159765753, 6.770226004962758, 7.747204703685402, 8.023706715763651, 8.367280496781873, 8.786388303944274, 9.760397800756041, 10.28893999839617, 10.75970285073443, 11.21633637958721, 11.75650394870226, 12.04862307280234, 12.85013057072269, 13.37578856334083, 13.80767707293004, 14.32464719432136, 14.82569896275900
