L(s) = 1 | + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 35-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 35-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8256783936 - 0.3850201581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8256783936 - 0.3850201581i\) |
\(L(1)\) |
\(\approx\) |
\(0.9657295487 - 0.2418462325i\) |
\(L(1)\) |
\(\approx\) |
\(0.9657295487 - 0.2418462325i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 - T \) |
| 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
−31.47365284387212510527484027178, −30.76198118182756399963417571918, −29.86383860276153121854188414741, −28.29821303154952302982613649817, −27.5523666212794504881485136308, −26.36871616010120181541707939936, −25.26505887346781658518871709209, −24.20369078163529232283781407910, −22.74636448527918262607168535777, −22.18109369445376156327064564481, −20.683856857880770865253802663443, −19.55003745177711351360128864969, −18.295376777617149728141500040726, −17.579108694749484804602030326419, −15.58456159252951795433077305233, −15.118598841467356039227958215117, −13.706457356116864517578608271846, −12.083209241227550718772095990492, −11.22443392675669114780611968392, −9.78287036852542958536656612719, −8.27934023819982130331178300235, −7.04610367991045475940697044530, −5.56009399288233496114948471525, −3.84186622036355904314648850343, −2.24004668425538474525233672394,
1.280152197260028030800112306148, 3.74002024390602897682147946262, 4.860245271433104475227951032356, 6.65778239132726212866136137028, 8.13183863401421268741523653060, 9.14548552589160229265556507650, 10.910787912465851977167363821390, 11.85602292923144602115842235229, 13.35613524784961168721824939651, 14.30662366215205834290305926085, 16.0230132207712621025606396335, 16.69240992689120755000267012648, 18.02081219647978144864204048689, 19.503557143583352681058482142154, 20.34071796558502850726197051049, 21.41205749308946722549639921677, 22.83902782531965293166734826815, 24.10088757754853338482249620263, 24.495186011995033430334416316712, 26.32332905636736910326283304543, 27.08266425690759396433972013943, 28.23440745442363445517025022235, 29.24257735426601907009727017624, 30.52325296682130191117812107755, 31.41750544205144077923310704548