L(s) = 1 | + (−0.866 − 0.5i)7-s + (−0.406 − 0.913i)11-s + (−0.913 − 0.406i)13-s + (−0.951 − 0.309i)17-s + (−0.951 − 0.309i)19-s + (0.994 + 0.104i)23-s + (−0.207 + 0.978i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (−0.5 + 0.866i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (−0.406 + 0.913i)59-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)7-s + (−0.406 − 0.913i)11-s + (−0.913 − 0.406i)13-s + (−0.951 − 0.309i)17-s + (−0.951 − 0.309i)19-s + (0.994 + 0.104i)23-s + (−0.207 + 0.978i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (−0.5 + 0.866i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (−0.406 + 0.913i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5015245284 - 0.1116445732i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5015245284 - 0.1116445732i\) |
\(L(1)\) |
\(\approx\) |
\(0.6810910422 - 0.08840069322i\) |
\(L(1)\) |
\(\approx\) |
\(0.6810910422 - 0.08840069322i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.406 - 0.913i)T \) |
| 13 | \( 1 + (-0.913 - 0.406i)T \) |
| 17 | \( 1 + (-0.951 - 0.309i)T \) |
| 19 | \( 1 + (-0.951 - 0.309i)T \) |
| 23 | \( 1 + (0.994 + 0.104i)T \) |
| 29 | \( 1 + (-0.207 + 0.978i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (-0.913 - 0.406i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.207 + 0.978i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.406 + 0.913i)T \) |
| 61 | \( 1 + (0.406 + 0.913i)T \) |
| 67 | \( 1 + (-0.978 + 0.207i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.587 + 0.809i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.669 - 0.743i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (-0.207 + 0.978i)T \) |
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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
−18.6624291218061700026121176085, −17.92361690973891111867169746898, −17.01481848338432054552170454801, −16.72073220199151704725701879735, −15.692140287473467186850495653174, −14.99147082423202460816388174482, −14.842421302120082475235568599, −13.49326851897106219930166229137, −13.02130564834750485541850086523, −12.43255321175400747217425267067, −11.720498495723228388222469086904, −10.88352457852789291588415141274, −10.006424778920325324560883980089, −9.58009970830349261546006822202, −8.77912160305172291433368239715, −8.04122276383154894783882103680, −6.95411994761541842470940096439, −6.70370549339100474784770456767, −5.69917559894178064824871425992, −4.87813604451789550930963265804, −4.20109292547293424911750102880, −3.24791361057984321561049548265, −2.28350800314305192061867505413, −1.89736912417662096499583223888, −0.2471760745458672908441357638,
0.26008140790681542984312877747, 1.31739401194756605181006205260, 2.61348083261809696883450136345, 3.01633442404222408546965351318, 4.00209907143010119610629991054, 4.80413138076480766405316093203, 5.61358236423708532795633063330, 6.439570551341206392226509897114, 7.118945307428962645093885458621, 7.73375862242666868564330576123, 8.89004571836513278621682216736, 9.14765231492282267714543221710, 10.24966436746022946430083228972, 10.73686117464642578034686751657, 11.37176700886489673467661970938, 12.4270494587492466840138459644, 13.08701441918426608452994333809, 13.38311118015633989750647537716, 14.4042012891631077253855803606, 15.03938325122608015689581513864, 15.78728981163786419119977415205, 16.52491221131880619744785992965, 16.93178814823672472470558635364, 17.801776102974703609066009636324, 18.48667506108926916425469322422