L(s) = 1 | + (−0.848 − 0.529i)2-s + (0.438 + 0.898i)4-s + (−0.766 − 0.642i)5-s + (0.961 − 0.275i)7-s + (0.104 − 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.719 − 0.694i)11-s + (0.882 + 0.469i)13-s + (−0.961 − 0.275i)14-s + (−0.615 + 0.788i)16-s + (−0.104 + 0.994i)17-s + (0.309 + 0.951i)19-s + (0.241 − 0.970i)20-s + (0.241 + 0.970i)22-s + (−0.719 + 0.694i)23-s + ⋯ |
L(s) = 1 | + (−0.848 − 0.529i)2-s + (0.438 + 0.898i)4-s + (−0.766 − 0.642i)5-s + (0.961 − 0.275i)7-s + (0.104 − 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.719 − 0.694i)11-s + (0.882 + 0.469i)13-s + (−0.961 − 0.275i)14-s + (−0.615 + 0.788i)16-s + (−0.104 + 0.994i)17-s + (0.309 + 0.951i)19-s + (0.241 − 0.970i)20-s + (0.241 + 0.970i)22-s + (−0.719 + 0.694i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7863720711 + 0.08820830014i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7863720711 + 0.08820830014i\) |
\(L(1)\) |
\(\approx\) |
\(0.6855993707 - 0.1163097581i\) |
\(L(1)\) |
\(\approx\) |
\(0.6855993707 - 0.1163097581i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.848 - 0.529i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.961 - 0.275i)T \) |
| 11 | \( 1 + (-0.719 - 0.694i)T \) |
| 13 | \( 1 + (0.882 + 0.469i)T \) |
| 17 | \( 1 + (-0.104 + 0.994i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| 23 | \( 1 + (-0.719 + 0.694i)T \) |
| 29 | \( 1 + (0.848 + 0.529i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.990 + 0.139i)T \) |
| 43 | \( 1 + (-0.848 - 0.529i)T \) |
| 47 | \( 1 + (-0.990 - 0.139i)T \) |
| 53 | \( 1 + (-0.104 + 0.994i)T \) |
| 59 | \( 1 + (0.882 + 0.469i)T \) |
| 61 | \( 1 + (-0.173 - 0.984i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.104 + 0.994i)T \) |
| 79 | \( 1 + (-0.438 + 0.898i)T \) |
| 83 | \( 1 + (0.0348 - 0.999i)T \) |
| 89 | \( 1 + (0.913 + 0.406i)T \) |
| 97 | \( 1 + (0.961 - 0.275i)T \) |
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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
−22.33150021324901415621168246852, −21.03581084871023953944621669022, −20.354956210813481797979460749811, −19.66332436037683844311582492847, −18.61176517294238962284403153470, −17.977350824219971041359792849489, −17.78439271950360071941303408742, −16.27618164630003255568183146667, −15.73468066348378687041039736372, −15.06927765302778893262339237069, −14.373397129692159430611401807725, −13.38722079392914232367870453554, −11.94204159325272796216245735283, −11.33098974417725910811910358975, −10.602403241489562524698561567753, −9.777283408543379893564166478669, −8.552204002058169895350631852514, −8.03140040884107681158026244257, −7.256651030482703402590509023081, −6.43748637305933266345481009310, −5.24655913091742574493044525305, −4.49844906952304854450080987289, −2.93121302014149527519341700529, −2.03956095655355139909748517767, −0.56457728082755578182290633615,
1.09449875633754230811722436798, 1.79875668023087529234033357607, 3.364608666434146163117359499700, 3.99949366897332480794720142043, 5.12013943821321705287895304263, 6.373841674621506657832227580708, 7.62950264326690824308621318415, 8.319563697316784394976394569966, 8.56026818076003456787022718487, 9.92540962442903218285398305323, 10.770971563917869510094484706894, 11.46960029949989963082027087128, 12.08804215714227303574059261511, 13.09524254994082743976998040846, 13.89825151941810403487462409743, 15.19626848091721395174495955633, 15.98982368180556735778625122278, 16.63142145350261504311787338882, 17.39179164922180716801782290770, 18.41611662380788374735576014829, 18.82361087716702819716715115847, 19.95894678154652620141596035726, 20.35145796834261177399716032759, 21.31857198210005320984682883302, 21.63041874944649309424304815406