| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 11-s − 2·15-s + 5·17-s − 2·19-s − 8·21-s + 6·23-s + 25-s − 4·27-s + 31-s − 2·33-s + 4·35-s − 2·37-s − 2·41-s + 43-s − 45-s + 9·49-s + 10·51-s − 6·53-s + 55-s − 4·57-s − 7·59-s + 4·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.516·15-s + 1.21·17-s − 0.458·19-s − 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.179·31-s − 0.348·33-s + 0.676·35-s − 0.328·37-s − 0.312·41-s + 0.152·43-s − 0.149·45-s + 9/7·49-s + 1.40·51-s − 0.824·53-s + 0.134·55-s − 0.529·57-s − 0.911·59-s + 0.512·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
| 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
−14.15814266720392, −14.03518727974968, −13.17789617115861, −12.92008955448262, −12.58088202832985, −11.91560793477563, −11.34892427949500, −10.71014015458692, −10.12869309906353, −9.710588792225171, −9.275864868062495, −8.681794422118543, −8.348584819850628, −7.645756950824508, −7.262566949701155, −6.690395474277741, −6.073382635890435, −5.499440755895265, −4.791134528377399, −4.021261402660905, −3.426055718275643, −3.099205126011570, −2.682830350666998, −1.822202527019703, −0.8624316996876890, 0,
0.8624316996876890, 1.822202527019703, 2.682830350666998, 3.099205126011570, 3.426055718275643, 4.021261402660905, 4.791134528377399, 5.499440755895265, 6.073382635890435, 6.690395474277741, 7.262566949701155, 7.645756950824508, 8.348584819850628, 8.681794422118543, 9.275864868062495, 9.710588792225171, 10.12869309906353, 10.71014015458692, 11.34892427949500, 11.91560793477563, 12.58088202832985, 12.92008955448262, 13.17789617115861, 14.03518727974968, 14.15814266720392
