| L(s) = 1 | + 2·5-s + 2·7-s − 4·11-s − 4·13-s − 2·17-s + 4·23-s − 25-s − 4·31-s + 4·35-s + 2·37-s + 12·43-s + 2·47-s − 3·49-s − 4·53-s − 8·55-s − 4·59-s + 10·61-s − 8·65-s − 8·67-s + 10·71-s − 2·73-s − 8·77-s − 14·79-s − 12·83-s − 4·85-s + 10·89-s − 8·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 1.20·11-s − 1.10·13-s − 0.485·17-s + 0.834·23-s − 1/5·25-s − 0.718·31-s + 0.676·35-s + 0.328·37-s + 1.82·43-s + 0.291·47-s − 3/7·49-s − 0.549·53-s − 1.07·55-s − 0.520·59-s + 1.28·61-s − 0.992·65-s − 0.977·67-s + 1.18·71-s − 0.234·73-s − 0.911·77-s − 1.57·79-s − 1.31·83-s − 0.433·85-s + 1.05·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 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 |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.05189229738289, −12.78535349150175, −12.29541855662595, −11.66816078997884, −11.09924921177065, −10.91608406919534, −10.22372775932669, −9.975777436676666, −9.379434573686756, −8.984113478334215, −8.474578710161864, −7.804803962210787, −7.496512483397093, −7.115149908749503, −6.364497869629610, −5.862379570902125, −5.374672872005797, −4.987437894206872, −4.555287276830529, −3.946775901846484, −3.055713251702345, −2.593445191995638, −2.141680315642242, −1.644694231238709, −0.7943637587232874, 0,
0.7943637587232874, 1.644694231238709, 2.141680315642242, 2.593445191995638, 3.055713251702345, 3.946775901846484, 4.555287276830529, 4.987437894206872, 5.374672872005797, 5.862379570902125, 6.364497869629610, 7.115149908749503, 7.496512483397093, 7.804803962210787, 8.474578710161864, 8.984113478334215, 9.379434573686756, 9.975777436676666, 10.22372775932669, 10.91608406919534, 11.09924921177065, 11.66816078997884, 12.29541855662595, 12.78535349150175, 13.05189229738289
