| L(s) = 1 | − 3·5-s − 7-s + 5·11-s − 2·13-s + 2·17-s + 6·19-s + 5·23-s + 4·25-s + 2·31-s + 3·35-s − 5·37-s − 5·41-s − 10·43-s − 11·47-s − 6·49-s − 9·53-s − 15·55-s + 10·61-s + 6·65-s + 2·67-s + 2·71-s − 7·73-s − 5·77-s + 12·79-s − 15·83-s − 6·85-s + 14·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.50·11-s − 0.554·13-s + 0.485·17-s + 1.37·19-s + 1.04·23-s + 4/5·25-s + 0.359·31-s + 0.507·35-s − 0.821·37-s − 0.780·41-s − 1.52·43-s − 1.60·47-s − 6/7·49-s − 1.23·53-s − 2.02·55-s + 1.28·61-s + 0.744·65-s + 0.244·67-s + 0.237·71-s − 0.819·73-s − 0.569·77-s + 1.35·79-s − 1.64·83-s − 0.650·85-s + 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.00126215366175, −12.68881808245071, −12.01008509206589, −11.81815076712832, −11.39955416388731, −11.17420632604125, −10.18777701294824, −9.940013470686185, −9.440986885838939, −8.947227084680309, −8.393658700374603, −8.027448689654545, −7.387742069079863, −7.042901704421983, −6.627951874307423, −6.128571394758082, −5.290006524755715, −4.829425983520195, −4.506485702828951, −3.530325037330483, −3.417647787859215, −3.139590975409970, −2.010134017588977, −1.366105705764614, −0.7588172391682669, 0,
0.7588172391682669, 1.366105705764614, 2.010134017588977, 3.139590975409970, 3.417647787859215, 3.530325037330483, 4.506485702828951, 4.829425983520195, 5.290006524755715, 6.128571394758082, 6.627951874307423, 7.042901704421983, 7.387742069079863, 8.027448689654545, 8.393658700374603, 8.947227084680309, 9.440986885838939, 9.940013470686185, 10.18777701294824, 11.17420632604125, 11.39955416388731, 11.81815076712832, 12.01008509206589, 12.68881808245071, 13.00126215366175
