L(s) = 1 | + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (−0.580 − 0.814i)5-s + (−0.0475 + 0.998i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (−0.235 + 0.971i)13-s + (−0.580 − 0.814i)14-s + (−0.888 − 0.458i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.928 + 0.371i)20-s + (−0.0475 − 0.998i)23-s + (−0.327 + 0.945i)25-s + (−0.415 − 0.909i)26-s + ⋯ |
L(s) = 1 | + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (−0.580 − 0.814i)5-s + (−0.0475 + 0.998i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (−0.235 + 0.971i)13-s + (−0.580 − 0.814i)14-s + (−0.888 − 0.458i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.928 + 0.371i)20-s + (−0.0475 − 0.998i)23-s + (−0.327 + 0.945i)25-s + (−0.415 − 0.909i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7539004827 + 0.002174889612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7539004827 + 0.002174889612i\) |
\(L(1)\) |
\(\approx\) |
\(0.6490935096 + 0.09363325691i\) |
\(L(1)\) |
\(\approx\) |
\(0.6490935096 + 0.09363325691i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.786 + 0.618i)T \) |
| 5 | \( 1 + (-0.580 - 0.814i)T \) |
| 7 | \( 1 + (-0.0475 + 0.998i)T \) |
| 13 | \( 1 + (-0.235 + 0.971i)T \) |
| 17 | \( 1 + (-0.654 - 0.755i)T \) |
| 19 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-0.0475 - 0.998i)T \) |
| 29 | \( 1 + (0.981 + 0.189i)T \) |
| 31 | \( 1 + (0.723 - 0.690i)T \) |
| 37 | \( 1 + (-0.959 - 0.281i)T \) |
| 41 | \( 1 + (0.928 + 0.371i)T \) |
| 43 | \( 1 + (-0.580 + 0.814i)T \) |
| 47 | \( 1 + (-0.928 + 0.371i)T \) |
| 53 | \( 1 + (-0.841 - 0.540i)T \) |
| 59 | \( 1 + (0.786 + 0.618i)T \) |
| 61 | \( 1 + (-0.928 + 0.371i)T \) |
| 67 | \( 1 + (0.928 + 0.371i)T \) |
| 71 | \( 1 + (0.654 - 0.755i)T \) |
| 73 | \( 1 + (-0.841 + 0.540i)T \) |
| 79 | \( 1 + (0.995 - 0.0950i)T \) |
| 83 | \( 1 + (0.0475 - 0.998i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (0.580 - 0.814i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.30027965502275509581321537424, −20.4183705821858158728759805650, −19.69290223928216975198734669760, −19.35912928207180783487242075992, −18.33934911109327385989750445401, −17.59238331817649415903187028900, −17.128717287573152841467259957758, −15.91581049930870360004896203863, −15.503051889553464143614156239858, −14.2830811829812645207134988124, −13.51529980730175156129401313579, −12.5424258305363834628453455780, −11.77802438475594810055795924259, −10.90024105150511835205402749750, −10.35148414595023580521903618450, −9.778854277028067013165763550448, −8.42547119820896677414244994564, −7.82836545655352162749938338809, −7.12104226218595390582035294716, −6.30040114942913371141680806421, −4.74937883077123396569001746952, −3.59703019690043181367635242126, −3.269442201015397332746584263759, −1.98691701323568125597626813377, −0.79605575293329713490887027123,
0.59569240924526805168951192805, 1.86341866142403355313786606007, 2.85262888532170898097209825774, 4.60528591599302553536847127991, 4.93444835276303531206959366306, 6.146074093178964490017342046777, 6.89385935173626032027923142475, 7.87051081291614344780862667625, 8.70581812855824942348818322177, 9.15536220244146939101828761343, 9.929253943635576510857312117321, 11.32171092886157151854718798607, 11.68288314307527620285780787063, 12.67201847960940296174683536320, 13.74777640284562086016344468284, 14.61226909711810928048138653707, 15.51624489396546430243310348726, 16.02975390606777874678299812117, 16.582299374253123465482151998351, 17.6321189084588483408277413556, 18.2211140589420551664800128303, 19.173275553474825516428038200104, 19.5890049190861183547858889894, 20.513202485327435222276461341, 21.257558992537592647342808717970