L(s) = 1 | + (0.981 − 0.189i)2-s + (−0.5 − 0.866i)3-s + (0.928 − 0.371i)4-s + (−0.888 + 0.458i)5-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 + 0.866i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.723 − 0.690i)16-s + (0.580 − 0.814i)17-s + (−0.327 + 0.945i)18-s + (0.580 + 0.814i)19-s + (−0.654 + 0.755i)20-s + ⋯ |
L(s) = 1 | + (0.981 − 0.189i)2-s + (−0.5 − 0.866i)3-s + (0.928 − 0.371i)4-s + (−0.888 + 0.458i)5-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 + 0.866i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.723 − 0.690i)16-s + (0.580 − 0.814i)17-s + (−0.327 + 0.945i)18-s + (0.580 + 0.814i)19-s + (−0.654 + 0.755i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.787286424 - 1.009559464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.787286424 - 1.009559464i\) |
\(L(1)\) |
\(\approx\) |
\(1.423404642 - 0.4838115969i\) |
\(L(1)\) |
\(\approx\) |
\(1.423404642 - 0.4838115969i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.981 - 0.189i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.888 + 0.458i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (0.580 - 0.814i)T \) |
| 19 | \( 1 + (0.580 + 0.814i)T \) |
| 23 | \( 1 + (0.723 - 0.690i)T \) |
| 29 | \( 1 + (0.415 - 0.909i)T \) |
| 31 | \( 1 + (-0.786 + 0.618i)T \) |
| 37 | \( 1 + (0.928 + 0.371i)T \) |
| 41 | \( 1 + (-0.654 - 0.755i)T \) |
| 43 | \( 1 + (0.841 - 0.540i)T \) |
| 47 | \( 1 + (-0.327 - 0.945i)T \) |
| 53 | \( 1 + (0.723 + 0.690i)T \) |
| 59 | \( 1 + (0.981 + 0.189i)T \) |
| 61 | \( 1 + (-0.327 - 0.945i)T \) |
| 67 | \( 1 + (-0.327 + 0.945i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.235 + 0.971i)T \) |
| 79 | \( 1 + (-0.888 + 0.458i)T \) |
| 83 | \( 1 + (-0.959 + 0.281i)T \) |
| 89 | \( 1 + (-0.995 - 0.0950i)T \) |
| 97 | \( 1 + (0.841 - 0.540i)T \) |
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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
−22.34400987341438172933739183060, −21.53982725877902863689181314655, −20.84542835611368306824203046087, −20.005570597743677176432873796723, −19.55370206476427405796911954902, −17.99844695310945178697791948253, −17.07598335607251057012124486513, −16.40621038410651019477078784956, −15.71387978855186288682200286935, −15.069626400849233758983648814, −14.50787046603382893494720734198, −13.07112304017745976560288107483, −12.59830295277110220648243119877, −11.573912093129009083044895486622, −11.11718544289602624850581745733, −10.17038005827450295913456581298, −9.00481937411079533360039886951, −7.96412013667749426191369351438, −7.16429697736940511746375023534, −5.935105304276220750101074589466, −5.222082605526967094006579492604, −4.52072949388596831737319556864, −3.57650162650540694283767030365, −2.96355874818923892122377445266, −1.06378932130890352918696190319,
0.90667488125107485977089004429, 2.15811621443513919666945421101, 3.07335350931863783159373546884, 4.128572026330893554378782106199, 5.06917610407374261200746634712, 6.00061470419916843706680452016, 7.02366126846845911371398165500, 7.31530615351248697820577570549, 8.44082441482486939885861935815, 9.98275199056455558475826440985, 10.93730703423074254843718862089, 11.708365669590818272169488248109, 12.06555284124670034313180411499, 12.92865653666940313762224479740, 14.02403498188913823791012175889, 14.36519299880560722902497001099, 15.48046345709048812468654325127, 16.36103650063710240391968543699, 16.887153309171958385348523171, 18.42293907184566254603572665515, 18.776355616788209954776185812693, 19.59041338496849822223497850782, 20.36120896932754034120303371023, 21.31434907455235514632603718749, 22.307448983725995075296788340673