L(s) = 1 | + (0.988 − 0.149i)3-s + (−0.623 − 0.781i)5-s + (0.955 − 0.294i)9-s + (0.733 − 0.680i)11-s + (0.955 + 0.294i)13-s + (−0.733 − 0.680i)15-s + (−0.826 + 0.563i)17-s + (−0.826 − 0.563i)23-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + (−0.826 + 0.563i)29-s + (0.5 + 0.866i)31-s + (0.623 − 0.781i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)39-s + ⋯ |
L(s) = 1 | + (0.988 − 0.149i)3-s + (−0.623 − 0.781i)5-s + (0.955 − 0.294i)9-s + (0.733 − 0.680i)11-s + (0.955 + 0.294i)13-s + (−0.733 − 0.680i)15-s + (−0.826 + 0.563i)17-s + (−0.826 − 0.563i)23-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + (−0.826 + 0.563i)29-s + (0.5 + 0.866i)31-s + (0.623 − 0.781i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0444 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0444 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.306574342 - 2.206229460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.306574342 - 2.206229460i\) |
\(L(1)\) |
\(\approx\) |
\(1.406164741 - 0.3783943673i\) |
\(L(1)\) |
\(\approx\) |
\(1.406164741 - 0.3783943673i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.988 - 0.149i)T \) |
| 5 | \( 1 + (-0.623 - 0.781i)T \) |
| 11 | \( 1 + (0.733 - 0.680i)T \) |
| 13 | \( 1 + (0.955 + 0.294i)T \) |
| 17 | \( 1 + (-0.826 + 0.563i)T \) |
| 23 | \( 1 + (-0.826 - 0.563i)T \) |
| 29 | \( 1 + (-0.826 + 0.563i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.826 + 0.563i)T \) |
| 41 | \( 1 + (0.365 - 0.930i)T \) |
| 43 | \( 1 + (0.988 + 0.149i)T \) |
| 47 | \( 1 + (-0.733 + 0.680i)T \) |
| 53 | \( 1 + (0.900 - 0.433i)T \) |
| 59 | \( 1 + (-0.365 - 0.930i)T \) |
| 61 | \( 1 + (-0.0747 - 0.997i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.826 + 0.563i)T \) |
| 73 | \( 1 + (0.733 + 0.680i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.222 + 0.974i)T \) |
| 89 | \( 1 + (0.955 - 0.294i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−18.59672690210830517488713452893, −18.16157019078183902204134129641, −17.408420607559104901037876935438, −16.271501606661776071225096443630, −15.73156457737070288557918412612, −15.090971472054769110641258878176, −14.70426148914860744552648172146, −13.703941311600300990246350402487, −13.474262790391332047869987774057, −12.37973180707946980336145965895, −11.65993082674128537621644481310, −10.9684219488434393820630428756, −10.214153630216464705495014704602, −9.45688504268425114935756786530, −8.86858558570871520873281815467, −7.95399460289855273667121270874, −7.49726452240239146244740822564, −6.72251972301284862299038545894, −6.01503167506369638499003599810, −4.74739741337867074670060511429, −3.868988770309844078568286284747, −3.68432874345184757916994098832, −2.538720400316453502162493595334, −1.99825801235875188468224911402, −0.8369322112175878882811420649,
0.47822285055385519701505983403, 1.3696858768023602959320758293, 2.01724033202533403424741966550, 3.2368008386532543456200105285, 3.85816473703608392017782966442, 4.312537191600673120603733006989, 5.37209651528801365225256195072, 6.42539438879163318720266229125, 6.94471471602778409709850870042, 8.108046076689637474005400133521, 8.398738903733450480925031422426, 9.00099994151946108826306565619, 9.6242240557093197192448164927, 10.77188513703159115529682364034, 11.30687655232421241436927908662, 12.34321255285476657614969130765, 12.68879182871788516099022627715, 13.636873604568465793896017128217, 14.0335297274692406220268748479, 14.82816641439925268776075117313, 15.76048059423482889834371394657, 15.94630001698360108314733836509, 16.82874402442189622727098809578, 17.61870131007499154008828176766, 18.51323584613529377795247145738