| L(s) = 1 | + (0.781 − 1.17i)2-s + (1.69 + 0.359i)3-s + (−0.777 − 1.84i)4-s + (1.22 − 3.37i)5-s + (1.74 − 1.71i)6-s + (−0.984 − 0.173i)7-s + (−2.77 − 0.524i)8-s + (2.74 + 1.21i)9-s + (−3.01 − 4.08i)10-s + (−4.02 + 1.46i)11-s + (−0.654 − 3.40i)12-s + (1.16 − 0.975i)13-s + (−0.974 + 1.02i)14-s + (3.29 − 5.27i)15-s + (−2.79 + 2.86i)16-s + (2.93 − 1.69i)17-s + ⋯ |
| L(s) = 1 | + (0.552 − 0.833i)2-s + (0.978 + 0.207i)3-s + (−0.388 − 0.921i)4-s + (0.548 − 1.50i)5-s + (0.713 − 0.700i)6-s + (−0.372 − 0.0656i)7-s + (−0.982 − 0.185i)8-s + (0.913 + 0.405i)9-s + (−0.952 − 1.29i)10-s + (−1.21 + 0.441i)11-s + (−0.189 − 0.981i)12-s + (0.322 − 0.270i)13-s + (−0.260 + 0.273i)14-s + (0.849 − 1.36i)15-s + (−0.697 + 0.716i)16-s + (0.711 − 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20788 - 2.38998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20788 - 2.38998i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.781 + 1.17i)T \) |
| 3 | \( 1 + (-1.69 - 0.359i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (-1.22 + 3.37i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (4.02 - 1.46i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.16 + 0.975i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.93 + 1.69i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.42 - 0.822i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.881 + 5.00i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.95 - 4.70i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.28 + 1.10i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.36 - 9.29i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.64 - 1.95i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.34 - 6.45i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.809 + 4.59i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 3.78iT - 53T^{2} \) |
| 59 | \( 1 + (4.92 + 1.79i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.72 + 9.80i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.74 + 4.46i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.44 + 5.96i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.10 - 5.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.47 + 5.33i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.52 - 7.15i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.728 + 0.420i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.7 - 3.89i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.886352915371077670501296593954, −9.473570762391240783546238466064, −8.524420503615455246924208293701, −7.81548620008471936662008707961, −6.21691881144458287835870213887, −5.05586769556777719398904566624, −4.65351240820699392657351325062, −3.33895619123728509365034773763, −2.34737750197663880607627971034, −1.09334662793700094071760555929,
2.44000077415286932804267527757, 3.11548015093293956153905224221, 4.00603308912049552782027753203, 5.67755906938637055913940309360, 6.20777570129463433414009098399, 7.45388319301953919415964469088, 7.56361991922724459178018117835, 8.788636229261113352151768192116, 9.699287500060082833831187919753, 10.44552681664125619680829313509
