L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)28-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)28-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04036695815 + 0.03985259632i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04036695815 + 0.03985259632i\) |
\(L(1)\) |
\(\approx\) |
\(0.7734993111 - 0.4599048924i\) |
\(L(1)\) |
\(\approx\) |
\(0.7734993111 - 0.4599048924i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 - T \) |
| 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 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 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
−26.19445502023629725750375969926, −25.65862329264384950340240202931, −24.38783797566834859471827877901, −24.01219742140349823298829853774, −22.689775705668213677554681390246, −21.92454617415189911077530507215, −21.26356800135504599485052193284, −19.799342208739800791900469680717, −18.59141850197650169480036127370, −17.82622539279848824922934023044, −16.53646397253016858367430525879, −15.83431449570395561589178808742, −14.98217375703875362218300565129, −13.8518911171774630114493259050, −12.916864194716775659839329157224, −12.06491145104062307871126041836, −10.687337706531032134333893274160, −9.067634347979663950543442975022, −8.44938752208801562682725370047, −7.07274065591206591782673410919, −6.01873887717105346198091711098, −5.17743418041809450499925381689, −3.71397524789348630450427302332, −2.56131871147668518363280159347, −0.01599015317802703654715240319,
1.56099968771323992192225360619, 2.98158282189680574977783247169, 4.11047744722598572818375235011, 5.173321288319805652262028985987, 6.54497083420190993477097233400, 7.80509673488507874449194691664, 9.53393641390114402369254317917, 10.07605993014298697161986098281, 11.23623877534812642893270220483, 12.265360012985461053283061996539, 13.279554618710459860862269681602, 13.97816844547600938687113976085, 15.180357385944424282852557790889, 16.237316266939445392295648745965, 17.635314728581786437351751275275, 18.47825752970611490146948230203, 19.689755230744162388815703318848, 20.30331921125632426642708166178, 21.14396190338881492112056662922, 22.39645030180865818857101755815, 23.01474038993171844684018127757, 23.806289074991558832496463727561, 25.01845580894510599260889312287, 26.23041867278259689690019835793, 27.14351099242078852693584203610