L(s) = 1 | + (−0.5 + 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s − 29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)37-s + 41-s − i·43-s + (0.866 − 0.5i)47-s + (0.866 + 0.5i)53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s − 29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)37-s + 41-s − i·43-s + (0.866 − 0.5i)47-s + (0.866 + 0.5i)53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.419846908 + 0.7305671897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.419846908 + 0.7305671897i\) |
\(L(1)\) |
\(\approx\) |
\(1.008241375 + 0.06938860474i\) |
\(L(1)\) |
\(\approx\) |
\(1.008241375 + 0.06938860474i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 - iT \) |
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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
−21.676714630873111087229067193767, −21.201824146346498408494891655877, −20.20146052348008169346929200892, −19.28895878862874038463123216322, −18.79965913769418573554721665354, −17.73845767655963701703738606366, −17.077160166417015797947800315480, −16.07073539879259296455627597326, −15.53961454584672704678514377104, −14.48130911156022707050612841761, −13.625813834973149946741041285359, −13.06008290319100893382207912298, −11.91447371810810779406202837928, −11.13123421371425411249282466677, −10.49625350780570463218791416044, −9.13265990010577123089590769142, −8.82053853333936563915868914781, −7.54362823133393184554470289683, −6.79147753125764199467731983019, −5.77560030713564689251259545479, −4.87269655302776978355124240617, −3.84034674922606737265136423817, −2.82956882666799970971341922314, −1.744598842914779703693959745755, −0.446508001518676856018559203956,
0.8282237958255442250184918830, 2.155801764800821458324117342133, 3.04354433989887468491379419463, 4.23919368139397678041332573886, 5.13058390610598137304188425474, 6.006298471349133659361774940, 7.17408000309204316141571068917, 7.79100939316839587226876741098, 8.83991336743908020148738500401, 9.77150022067394092030166946551, 10.51719944967586521138770952815, 11.395241309161247912693448544664, 12.42101762055983111798974212829, 13.043879444537855062314886242, 13.90508705464900918845182278942, 15.036090129464693658087344993, 15.42260319944684787228149334645, 16.45964245795700618683114968809, 17.32216272309894643213170357974, 18.100184083181217522750778452288, 18.71741519401011045184665198910, 19.81833134945952778468114744806, 20.54315679283748287034894936132, 20.99927254360297143423649827621, 22.37032742356959338113096875667