| L(s) = 1 | + (0.303 − 2.30i)2-s + (0.0950 + 0.279i)3-s + (−3.27 − 0.878i)4-s + (−1.21 + 0.0798i)5-s + (0.673 − 0.133i)6-s + (−1.24 + 2.99i)8-s + (2.31 − 1.77i)9-s + (−0.185 + 2.82i)10-s + (−1.91 − 2.17i)11-s + (−0.0656 − 1.00i)12-s + (−0.992 + 0.992i)13-s + (−0.138 − 0.333i)15-s + (0.638 + 0.368i)16-s + (−4.03 + 0.857i)17-s + (−3.38 − 5.85i)18-s + (−3.24 − 0.427i)19-s + ⋯ |
| L(s) = 1 | + (0.214 − 1.62i)2-s + (0.0548 + 0.161i)3-s + (−1.63 − 0.439i)4-s + (−0.544 + 0.0356i)5-s + (0.274 − 0.0546i)6-s + (−0.438 + 1.05i)8-s + (0.770 − 0.591i)9-s + (−0.0586 + 0.894i)10-s + (−0.576 − 0.656i)11-s + (−0.0189 − 0.289i)12-s + (−0.275 + 0.275i)13-s + (−0.0356 − 0.0860i)15-s + (0.159 + 0.0921i)16-s + (−0.978 + 0.208i)17-s + (−0.797 − 1.38i)18-s + (−0.745 − 0.0981i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.377722 + 0.563919i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.377722 + 0.563919i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 + (4.03 - 0.857i)T \) |
| good | 2 | \( 1 + (-0.303 + 2.30i)T + (-1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (-0.0950 - 0.279i)T + (-2.38 + 1.82i)T^{2} \) |
| 5 | \( 1 + (1.21 - 0.0798i)T + (4.95 - 0.652i)T^{2} \) |
| 11 | \( 1 + (1.91 + 2.17i)T + (-1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.992 - 0.992i)T - 13iT^{2} \) |
| 19 | \( 1 + (3.24 + 0.427i)T + (18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (1.71 + 0.582i)T + (18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (1.94 + 2.90i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-1.64 + 0.557i)T + (24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (-0.462 - 0.405i)T + (4.82 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.22 + 6.31i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (2.34 + 0.969i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-2.98 - 11.1i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.32 - 4.85i)T + (13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (1.10 + 8.39i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (8.68 - 4.28i)T + (37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (3.61 - 2.08i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.14 + 15.8i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-6.08 + 12.3i)T + (-44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (3.83 - 11.2i)T + (-62.6 - 48.0i)T^{2} \) |
| 83 | \( 1 + (7.64 - 3.16i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.31 - 2.22i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (3.34 - 2.23i)T + (37.1 - 89.6i)T^{2} \) |
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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
−9.798056743754378310254026518651, −9.167550048113586458928251552097, −8.211404010377130881767763546950, −7.10391837619864659605911926098, −5.92991639549881734239061579828, −4.45862153059093383092485373507, −4.08099635814381767296692457492, −2.98289025000973601323587593672, −1.88786914160886799095323298572, −0.29626108723291981486881275982,
2.21867012550005396872280660467, 4.07640152122427868555579223636, 4.69919014922721249842152842793, 5.60502560692592840121400082300, 6.72354106731242850166039194307, 7.30901640855841667046641627624, 7.973589905043993373487132183462, 8.652871548760519798890320391612, 9.768626515110872470760979070183, 10.62122774958292489192521510487
