| L(s) = 1 | − 5-s − 7-s − 11-s − 3·17-s − 3·19-s − 23-s + 25-s − 7·29-s − 6·31-s + 35-s + 8·37-s + 2·41-s − 5·43-s − 6·47-s + 49-s − 3·53-s + 55-s + 5·59-s + 7·61-s + 6·67-s − 10·71-s + 12·73-s + 77-s + 16·79-s + 3·83-s + 3·85-s − 3·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.727·17-s − 0.688·19-s − 0.208·23-s + 1/5·25-s − 1.29·29-s − 1.07·31-s + 0.169·35-s + 1.31·37-s + 0.312·41-s − 0.762·43-s − 0.875·47-s + 1/7·49-s − 0.412·53-s + 0.134·55-s + 0.650·59-s + 0.896·61-s + 0.733·67-s − 1.18·71-s + 1.40·73-s + 0.113·77-s + 1.80·79-s + 0.329·83-s + 0.325·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| 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
−13.08544322914434, −12.83939327005009, −12.36988131111215, −11.72636205746497, −11.22448724350561, −11.02681958133067, −10.47455241170536, −9.863252730810869, −9.422066461118321, −9.067685020251649, −8.352256282587901, −8.099661195227407, −7.474151467981262, −7.047977897013444, −6.475277109619764, −6.101431806836434, −5.420950048770254, −4.970923175523336, −4.360870454810438, −3.763299716112026, −3.511815619977413, −2.628448243744140, −2.202296842142029, −1.562919977531778, −0.6042290674764350, 0,
0.6042290674764350, 1.562919977531778, 2.202296842142029, 2.628448243744140, 3.511815619977413, 3.763299716112026, 4.360870454810438, 4.970923175523336, 5.420950048770254, 6.101431806836434, 6.475277109619764, 7.047977897013444, 7.474151467981262, 8.099661195227407, 8.352256282587901, 9.067685020251649, 9.422066461118321, 9.863252730810869, 10.47455241170536, 11.02681958133067, 11.22448724350561, 11.72636205746497, 12.36988131111215, 12.83939327005009, 13.08544322914434
