L(s) = 1 | + (−0.995 − 0.0896i)2-s + (−0.963 − 0.266i)3-s + (0.983 + 0.178i)4-s + (−0.550 + 0.834i)5-s + (0.936 + 0.351i)6-s + (−0.963 − 0.266i)8-s + (0.858 + 0.512i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)12-s + (−0.995 − 0.0896i)13-s + (0.753 − 0.657i)15-s + (0.936 + 0.351i)16-s + (0.134 − 0.990i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.691 + 0.722i)20-s + ⋯ |
L(s) = 1 | + (−0.995 − 0.0896i)2-s + (−0.963 − 0.266i)3-s + (0.983 + 0.178i)4-s + (−0.550 + 0.834i)5-s + (0.936 + 0.351i)6-s + (−0.963 − 0.266i)8-s + (0.858 + 0.512i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)12-s + (−0.995 − 0.0896i)13-s + (0.753 − 0.657i)15-s + (0.936 + 0.351i)16-s + (0.134 − 0.990i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.691 + 0.722i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2319851664 - 0.1798366557i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2319851664 - 0.1798366557i\) |
\(L(1)\) |
\(\approx\) |
\(0.4085695636 + 0.01766595652i\) |
\(L(1)\) |
\(\approx\) |
\(0.4085695636 + 0.01766595652i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.995 - 0.0896i)T \) |
| 3 | \( 1 + (-0.963 - 0.266i)T \) |
| 5 | \( 1 + (-0.550 + 0.834i)T \) |
| 13 | \( 1 + (-0.995 - 0.0896i)T \) |
| 17 | \( 1 + (0.134 - 0.990i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.900 + 0.433i)T \) |
| 29 | \( 1 + (0.473 + 0.880i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.473 + 0.880i)T \) |
| 41 | \( 1 + (-0.963 - 0.266i)T \) |
| 43 | \( 1 + (0.623 - 0.781i)T \) |
| 47 | \( 1 + (0.753 + 0.657i)T \) |
| 53 | \( 1 + (0.134 + 0.990i)T \) |
| 59 | \( 1 + (-0.963 + 0.266i)T \) |
| 61 | \( 1 + (0.134 - 0.990i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.691 - 0.722i)T \) |
| 73 | \( 1 + (0.753 - 0.657i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.995 + 0.0896i)T \) |
| 89 | \( 1 + (-0.222 + 0.974i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−23.84785095550059123154309711256, −22.968003704678165201554954296923, −21.64492670393633698100534529274, −21.187958463038602975005390121320, −19.95410106741019011356822069700, −19.4696163808943284897344100787, −18.45716409723485036488151950523, −17.323853169372692682508136505897, −17.11218626939608477741224647704, −16.14939207873037113024797583027, −15.52971910061531292342450631606, −14.64188958972766543341392131134, −12.850221111995153772711489103193, −12.19713672109342326076672327727, −11.48765878921480444598461933572, −10.49679771832384504011415677508, −9.79023003447695453750250013719, −8.755934198458370768450418498957, −7.90626634319278921500361293956, −6.87507036547063325171882592566, −5.96167079259922739351256236107, −4.90148123374918556828491425442, −3.926324085925219852992363708757, −2.19618027904776817855033575802, −0.892355659355009985694463477649,
0.31981461865366712714137688957, 1.88513509090489729929636250546, 2.95745657223959415548386483403, 4.357330795153277109902489703156, 5.74307557241816990998989253292, 6.66630424136949950837179554085, 7.39509123948173630011123906220, 8.08090205291437103229524722525, 9.57518467242326552811249750030, 10.32706471930536520383146179189, 11.0408587652647891939490310431, 11.96249478818876584425767898084, 12.35319448253090080608023619445, 13.92189582620162395989175924684, 15.10108004037742834626065722095, 15.79675050601238804847302494325, 16.72633137764180430246614229456, 17.388504703031933749444736149842, 18.342386924023116515906559128716, 18.77835772247364798589316985786, 19.614037253464374677842547745679, 20.54275953886511755498869842422, 21.79578694307449234791068994917, 22.28558174699326953423840022036, 23.39255736907197425703433981772