L(s) = 1 | + (0.0747 + 0.997i)5-s + (0.365 + 0.930i)11-s + (0.988 + 0.149i)13-s + (−0.222 + 0.974i)17-s + 19-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (0.733 + 0.680i)29-s + (−0.5 − 0.866i)31-s + (−0.222 + 0.974i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.365 − 0.930i)47-s + (0.222 + 0.974i)53-s + (−0.900 + 0.433i)55-s + ⋯ |
L(s) = 1 | + (0.0747 + 0.997i)5-s + (0.365 + 0.930i)11-s + (0.988 + 0.149i)13-s + (−0.222 + 0.974i)17-s + 19-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (0.733 + 0.680i)29-s + (−0.5 − 0.866i)31-s + (−0.222 + 0.974i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.365 − 0.930i)47-s + (0.222 + 0.974i)53-s + (−0.900 + 0.433i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4219555901 + 2.013338078i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4219555901 + 2.013338078i\) |
\(L(1)\) |
\(\approx\) |
\(1.037695287 + 0.4898114736i\) |
\(L(1)\) |
\(\approx\) |
\(1.037695287 + 0.4898114736i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.0747 + 0.997i)T \) |
| 11 | \( 1 + (0.365 + 0.930i)T \) |
| 13 | \( 1 + (0.988 + 0.149i)T \) |
| 17 | \( 1 + (-0.222 + 0.974i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.733 + 0.680i)T \) |
| 29 | \( 1 + (0.733 + 0.680i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.222 + 0.974i)T \) |
| 41 | \( 1 + (0.0747 + 0.997i)T \) |
| 43 | \( 1 + (-0.0747 + 0.997i)T \) |
| 47 | \( 1 + (-0.365 - 0.930i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 + (-0.826 - 0.563i)T \) |
| 61 | \( 1 + (0.733 + 0.680i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (-0.623 - 0.781i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.988 - 0.149i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 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
−19.87757526967983062689587183330, −19.03046937692857193989523597270, −18.18969716765114787499693165813, −17.58403244956204510057325581599, −16.64174491558774184928012699975, −15.907747025046557265718533733108, −15.797261522402714686109734385130, −14.15353414990429452615227943867, −13.92243864046977614090337650919, −13.06403343299843694571384216683, −12.20914785503713655593333135724, −11.58103009488214439519508631394, −10.77158030533189743828871195915, −9.79251757318425569268467718728, −8.95392445620093357240114977973, −8.5045582041990089047641727770, −7.61887223061647091807735184036, −6.55720403965244250677682720935, −5.72159874023963181537319081613, −5.07693621639851975780667326894, −4.05404275600919569630150306977, −3.32344231069152761100102984176, −2.133166069141681076378318195706, −1.02297161313958828497820969557, −0.41360313561777075361253343322,
1.28350188646637346502105809162, 2.02847477656914131704175407626, 3.1848911674944352537729504459, 3.813570067675498538789725836563, 4.789611039017661961706389396191, 5.98629436299125548720351656648, 6.45905349823645720437460744332, 7.37821069527137733389627662142, 8.057656519471076753792411761478, 9.118060050975985369194979758725, 9.91975292838766483039853584184, 10.5291357967938747941382377034, 11.43819444396297237387160815301, 11.95296848481185481655471891862, 13.071349477347904691928493508122, 13.700973426815172735020230730860, 14.53864651143539761750239866369, 15.12570083795887604603187852455, 15.83067818689563391387600623579, 16.696357884161684290506301974980, 17.72723279044693665439026822043, 18.05507766163293898260812983755, 18.816087075421266889669387216550, 19.70994507072984307541690175848, 20.23618084057083177202628793418