| L(s) = 1 | + (1.16 + 0.798i)2-s + (0.0708 + 1.73i)3-s + (0.724 + 1.86i)4-s + (−1.41 + 3.88i)5-s + (−1.29 + 2.07i)6-s + (−0.984 − 0.173i)7-s + (−0.642 + 2.75i)8-s + (−2.98 + 0.245i)9-s + (−4.75 + 3.41i)10-s + (3.69 − 1.34i)11-s + (−3.17 + 1.38i)12-s + (4.45 − 3.74i)13-s + (−1.01 − 0.989i)14-s + (−6.83 − 2.17i)15-s + (−2.94 + 2.70i)16-s + (0.487 − 0.281i)17-s + ⋯ |
| L(s) = 1 | + (0.825 + 0.564i)2-s + (0.0408 + 0.999i)3-s + (0.362 + 0.931i)4-s + (−0.633 + 1.73i)5-s + (−0.530 + 0.847i)6-s + (−0.372 − 0.0656i)7-s + (−0.227 + 0.973i)8-s + (−0.996 + 0.0817i)9-s + (−1.50 + 1.07i)10-s + (1.11 − 0.405i)11-s + (−0.916 + 0.400i)12-s + (1.23 − 1.03i)13-s + (−0.270 − 0.264i)14-s + (−1.76 − 0.561i)15-s + (−0.737 + 0.675i)16-s + (0.118 − 0.0682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0940398 - 2.17563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0940398 - 2.17563i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 - 0.798i)T \) |
| 3 | \( 1 + (-0.0708 - 1.73i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (1.41 - 3.88i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-3.69 + 1.34i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.45 + 3.74i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.487 + 0.281i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.89 - 2.24i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.363 - 2.05i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.56 - 1.86i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.87 + 1.03i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (5.89 + 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.624 + 0.744i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.741 - 2.03i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.34 - 7.65i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.190iT - 53T^{2} \) |
| 59 | \( 1 + (-11.8 - 4.30i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.97 - 11.2i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.91 + 8.23i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (4.06 + 7.03i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.844 + 1.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.80 - 5.72i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.47 - 4.59i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.82 + 2.78i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.54 + 2.38i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.92793309818395003541016128703, −10.17033007238006489526101196892, −8.986174846990588915771871271148, −8.011563631340819502853944337439, −7.18037247805101897315754455416, −6.18850767699702957438752649441, −5.67748498923966417467069843245, −4.06283616426180503710289454743, −3.48620892667982237106589739506, −2.95558504561603861344304369669,
0.928958758532424716244092393585, 1.70741997661344328801507576716, 3.44209404246202066795069100175, 4.33770673668102148954288376965, 5.26846565152595058590892230721, 6.33883689777437396602966125390, 7.02597671148469976014873336960, 8.406558824012833845989515710462, 8.941767831914748251068549579701, 9.792642816798111876579772914301
