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
−20.23618084057083177202628793418, −19.70994507072984307541690175848, −18.816087075421266889669387216550, −18.05507766163293898260812983755, −17.72723279044693665439026822043, −16.696357884161684290506301974980, −15.83067818689563391387600623579, −15.12570083795887604603187852455, −14.53864651143539761750239866369, −13.700973426815172735020230730860, −13.071349477347904691928493508122, −11.95296848481185481655471891862, −11.43819444396297237387160815301, −10.5291357967938747941382377034, −9.91975292838766483039853584184, −9.118060050975985369194979758725, −8.057656519471076753792411761478, −7.37821069527137733389627662142, −6.45905349823645720437460744332, −5.98629436299125548720351656648, −4.789611039017661961706389396191, −3.813570067675498538789725836563, −3.1848911674944352537729504459, −2.02847477656914131704175407626, −1.28350188646637346502105809162,
0.41360313561777075361253343322, 1.02297161313958828497820969557, 2.133166069141681076378318195706, 3.32344231069152761100102984176, 4.05404275600919569630150306977, 5.07693621639851975780667326894, 5.72159874023963181537319081613, 6.55720403965244250677682720935, 7.61887223061647091807735184036, 8.5045582041990089047641727770, 8.95392445620093357240114977973, 9.79251757318425569268467718728, 10.77158030533189743828871195915, 11.58103009488214439519508631394, 12.20914785503713655593333135724, 13.06403343299843694571384216683, 13.92243864046977614090337650919, 14.15353414990429452615227943867, 15.797261522402714686109734385130, 15.907747025046557265718533733108, 16.64174491558774184928012699975, 17.58403244956204510057325581599, 18.18969716765114787499693165813, 19.03046937692857193989523597270, 19.87757526967983062689587183330