L(s) = 1 | + (0.543 − 0.839i)2-s + (−0.859 − 0.511i)3-s + (−0.409 − 0.912i)4-s + (−0.190 − 0.981i)5-s + (−0.896 + 0.443i)6-s + (−0.114 − 0.993i)7-s + (−0.988 − 0.152i)8-s + (0.477 + 0.878i)9-s + (−0.927 − 0.373i)10-s + (0.338 + 0.941i)11-s + (−0.114 + 0.993i)12-s + (0.264 − 0.964i)13-s + (−0.896 − 0.443i)14-s + (−0.338 + 0.941i)15-s + (−0.665 + 0.746i)16-s + (0.720 + 0.693i)17-s + ⋯ |
L(s) = 1 | + (0.543 − 0.839i)2-s + (−0.859 − 0.511i)3-s + (−0.409 − 0.912i)4-s + (−0.190 − 0.981i)5-s + (−0.896 + 0.443i)6-s + (−0.114 − 0.993i)7-s + (−0.988 − 0.152i)8-s + (0.477 + 0.878i)9-s + (−0.927 − 0.373i)10-s + (0.338 + 0.941i)11-s + (−0.114 + 0.993i)12-s + (0.264 − 0.964i)13-s + (−0.896 − 0.443i)14-s + (−0.338 + 0.941i)15-s + (−0.665 + 0.746i)16-s + (0.720 + 0.693i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4551532620 - 0.8776858444i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4551532620 - 0.8776858444i\) |
\(L(1)\) |
\(\approx\) |
\(0.4611059991 - 0.7673110415i\) |
\(L(1)\) |
\(\approx\) |
\(0.4611059991 - 0.7673110415i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (0.543 - 0.839i)T \) |
| 3 | \( 1 + (-0.859 - 0.511i)T \) |
| 5 | \( 1 + (-0.190 - 0.981i)T \) |
| 7 | \( 1 + (-0.114 - 0.993i)T \) |
| 11 | \( 1 + (0.338 + 0.941i)T \) |
| 13 | \( 1 + (0.264 - 0.964i)T \) |
| 17 | \( 1 + (0.720 + 0.693i)T \) |
| 19 | \( 1 + (-0.953 - 0.301i)T \) |
| 23 | \( 1 + (-0.997 + 0.0765i)T \) |
| 29 | \( 1 + (0.817 + 0.575i)T \) |
| 31 | \( 1 + (0.988 - 0.152i)T \) |
| 37 | \( 1 + (0.477 - 0.878i)T \) |
| 41 | \( 1 + (-0.543 - 0.839i)T \) |
| 43 | \( 1 + (-0.0383 - 0.999i)T \) |
| 47 | \( 1 + (-0.606 + 0.795i)T \) |
| 53 | \( 1 + (-0.606 - 0.795i)T \) |
| 59 | \( 1 + (-0.771 - 0.636i)T \) |
| 61 | \( 1 + (-0.973 - 0.227i)T \) |
| 67 | \( 1 + (0.665 - 0.746i)T \) |
| 71 | \( 1 + (0.114 - 0.993i)T \) |
| 73 | \( 1 + (0.927 + 0.373i)T \) |
| 79 | \( 1 + (0.409 + 0.912i)T \) |
| 89 | \( 1 + (-0.896 + 0.443i)T \) |
| 97 | \( 1 + (-0.896 - 0.443i)T \) |
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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
−31.62550725228256182068985018552, −30.32367489288042116962253716640, −29.33960462384942223986960622940, −27.8823594095739659188577533271, −26.92905302538214441797396386091, −26.04006754427309385734248951990, −24.82229958280900917072252075087, −23.58206964082758358753879465935, −22.80411940388388124479288640564, −21.72778522954091171476267967922, −21.36559506589287494154821645775, −18.89950997152329734967016410264, −18.12939844269677671880977144029, −16.743928245443195571129564334351, −15.89992873144920021956673542415, −14.92658627062568620725983224905, −13.87398879352626126018120126577, −12.08395999402862251741319438079, −11.451656590378328334683430785661, −9.75720091901720440352509365851, −8.29783102514468349085150112670, −6.50463139497754647049413004891, −6.02051655400350031873893812032, −4.44254404672995969780079614219, −3.08794753022772968542513205019,
0.450560503932339680259795567343, 1.649393736231175984032570928398, 3.98826886316306961533431510150, 5.01509795686634142941036604313, 6.36113444313782304178735696956, 8.02634826742450422815149872209, 9.91370694395436034552892149919, 10.83305658674484815601141496504, 12.28843247515702161187743689674, 12.73578955364553252461332191720, 13.897807706719633056910319732840, 15.56457877775484835420456543912, 17.00737317045454962383136500729, 17.81771739069568260124965638616, 19.400169143193147466139270946804, 20.12219841878495651360928671473, 21.242799673207801482647424836430, 22.61233711079774147790156334434, 23.39280218673882134850799191023, 24.049072638117147715186791989658, 25.386687805093675738070178521750, 27.45564596228053725197562660029, 27.99056040521872477786995090444, 28.92477787430880056934155667436, 30.06526887991341762301555650053