L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 − 0.984i)20-s + (−0.939 − 0.342i)22-s + (0.173 + 0.984i)23-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 − 0.984i)20-s + (−0.939 − 0.342i)22-s + (0.173 + 0.984i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8490271807 + 0.3372969165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8490271807 + 0.3372969165i\) |
\(L(1)\) |
\(\approx\) |
\(1.115760427 + 0.3412711769i\) |
\(L(1)\) |
\(\approx\) |
\(1.115760427 + 0.3412711769i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 7 | \( 1 + (0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.939 + 0.342i)T \) |
| 13 | \( 1 + (0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (0.766 + 0.642i)T \) |
| 31 | \( 1 + (0.173 + 0.984i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.173 - 0.984i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (-0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.173 - 0.984i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.766 + 0.642i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.939 + 0.342i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−38.05705172963086020231747861133, −36.761480619543136652806529917104, −34.96354881398418543688449087082, −33.89173526726370811178011254070, −32.27545631674212666868772665971, −31.16979786221864763223296122943, −30.51271429056067775742478794253, −28.74522149472364032448013532545, −27.91319882342413445836187070183, −26.249945738646348600298867042345, −24.32369897415381571503674604675, −23.38976133549368094409861490292, −22.03107949704272862435790170247, −20.913104195132228126122931810409, −19.322195944851260909865068085509, −18.45614389719719568044762940434, −15.81171109138441681210955252200, −14.93299902446899029176490644766, −13.18995949240848713693921055338, −11.787834728111295578053700052211, −10.72996026014505713935126654307, −8.558903351798499135324663408425, −6.29164786968591117600241502764, −4.44438357855947998471443120743, −2.65531742487320095547138102958,
3.58950910323227086925668585890, 5.03532086681002160373466118417, 7.15829115002224906226372528073, 8.27354959631020868239674358954, 10.90287068662359416861863466120, 12.510859478141777837115129966788, 13.76428114525284607594016530240, 15.40254334374809065819309872172, 16.34456156505821516520485918029, 17.893140159843779489600389111051, 20.03598160893936930548290354701, 21.0423512759974302105356328023, 23.095527271012988967278063217144, 23.414775503556971784434443983558, 24.93949048412501969405726267577, 26.36976147454807292943786649772, 27.499378897128436615294809322413, 29.43613779766801443137720978188, 30.78390994133351781706870369410, 31.66519302300073343520532810926, 32.98450940829158470999413590497, 34.02887553914528176727145001360, 35.402034786379264472974615654630, 36.274620460130378984339463788882, 38.37006695871833379169365588964