L(s) = 1 | + (−0.963 + 0.266i)2-s + (−0.393 + 0.919i)3-s + (0.858 − 0.512i)4-s + (0.309 − 0.951i)5-s + (0.134 − 0.990i)6-s + (0.0448 + 0.998i)7-s + (−0.691 + 0.722i)8-s + (−0.691 − 0.722i)9-s + (−0.0448 + 0.998i)10-s + (−0.983 − 0.178i)11-s + (0.134 + 0.990i)12-s + (−0.983 + 0.178i)13-s + (−0.309 − 0.951i)14-s + (0.753 + 0.657i)15-s + (0.473 − 0.880i)16-s + (0.809 − 0.587i)17-s + ⋯ |
L(s) = 1 | + (−0.963 + 0.266i)2-s + (−0.393 + 0.919i)3-s + (0.858 − 0.512i)4-s + (0.309 − 0.951i)5-s + (0.134 − 0.990i)6-s + (0.0448 + 0.998i)7-s + (−0.691 + 0.722i)8-s + (−0.691 − 0.722i)9-s + (−0.0448 + 0.998i)10-s + (−0.983 − 0.178i)11-s + (0.134 + 0.990i)12-s + (−0.983 + 0.178i)13-s + (−0.309 − 0.951i)14-s + (0.753 + 0.657i)15-s + (0.473 − 0.880i)16-s + (0.809 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0297 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0297 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2426264199 - 0.2499569139i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2426264199 - 0.2499569139i\) |
\(L(1)\) |
\(\approx\) |
\(0.5058265180 + 0.05972650241i\) |
\(L(1)\) |
\(\approx\) |
\(0.5058265180 + 0.05972650241i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (-0.963 + 0.266i)T \) |
| 3 | \( 1 + (-0.393 + 0.919i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.0448 + 0.998i)T \) |
| 11 | \( 1 + (-0.983 - 0.178i)T \) |
| 13 | \( 1 + (-0.983 + 0.178i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.753 - 0.657i)T \) |
| 23 | \( 1 + (-0.623 - 0.781i)T \) |
| 29 | \( 1 + (-0.995 - 0.0896i)T \) |
| 31 | \( 1 + (-0.473 - 0.880i)T \) |
| 37 | \( 1 + (0.623 - 0.781i)T \) |
| 41 | \( 1 + (0.900 + 0.433i)T \) |
| 43 | \( 1 + (-0.550 - 0.834i)T \) |
| 47 | \( 1 + (0.393 + 0.919i)T \) |
| 53 | \( 1 + (-0.858 - 0.512i)T \) |
| 59 | \( 1 + (-0.134 - 0.990i)T \) |
| 61 | \( 1 + (0.0448 - 0.998i)T \) |
| 67 | \( 1 + (-0.858 + 0.512i)T \) |
| 73 | \( 1 + (-0.963 + 0.266i)T \) |
| 79 | \( 1 + (-0.691 + 0.722i)T \) |
| 83 | \( 1 + (0.134 + 0.990i)T \) |
| 89 | \( 1 + (0.858 + 0.512i)T \) |
| 97 | \( 1 + (0.900 - 0.433i)T \) |
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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
−31.23022353261666281836795175526, −30.07826908646760919228800569444, −29.599400453176283118253700901496, −28.71274119981739278012702506081, −27.30329989642534141673433368653, −26.262772896620229301767670891993, −25.45274896667722867501328230918, −24.13066177108557830831137714988, −23.04831228510025136759162288462, −21.724188419596345977889647737781, −20.26238671429453640578184590977, −19.22082800253780036520263179001, −18.24613007518567477159552480681, −17.48245786919492611420519173359, −16.432820981494618221501147279446, −14.65318431793824665230476328431, −13.263780910975975918007599367303, −11.964741877307611888906599820683, −10.70393031969370548254577821843, −9.95557662396741283940384157150, −7.69119575884552530984710375155, −7.36406155532242172602249041575, −5.83413070519927345826115808387, −3.09134527357856677316655009427, −1.61449575157304473122858136064,
0.24122653773325117853817158712, 2.51526105615436962294602864728, 5.04603377487500421143821226836, 5.800803363315235559708537448097, 7.86822208401419909557290518194, 9.206563679038621373360455513856, 9.80085411385194653143633551992, 11.32619219119491897666908926702, 12.42308379375625248408574561907, 14.57228838079409132947285232056, 15.79143805940653917077893235348, 16.4355518884038620458317197718, 17.55877192031750504625530187856, 18.629300589664833661135024261361, 20.20798679618809404951334593202, 21.0038766301683685459667531359, 22.08939657142088545250152996535, 23.81866226438341308251741137866, 24.72885260686647999384873293419, 25.89752292243540634876089556620, 26.91264377903480667188122131478, 28.05913005310905198359706440341, 28.52959806526908401967502544330, 29.459257642008564472233141590813, 31.63829180529169368575645550439