L(s) = 1 | + (−0.946 − 0.322i)2-s + (0.853 − 0.520i)3-s + (0.791 + 0.611i)4-s + (−0.967 − 0.252i)5-s + (−0.976 + 0.217i)6-s + (−0.983 + 0.181i)7-s + (−0.551 − 0.833i)8-s + (0.457 − 0.889i)9-s + (0.833 + 0.551i)10-s + (−0.288 − 0.957i)11-s + (0.994 + 0.109i)12-s + (0.322 − 0.946i)13-s + (0.989 + 0.145i)14-s + (−0.957 + 0.288i)15-s + (0.252 + 0.967i)16-s + (−0.424 + 0.905i)17-s + ⋯ |
L(s) = 1 | + (−0.946 − 0.322i)2-s + (0.853 − 0.520i)3-s + (0.791 + 0.611i)4-s + (−0.967 − 0.252i)5-s + (−0.976 + 0.217i)6-s + (−0.983 + 0.181i)7-s + (−0.551 − 0.833i)8-s + (0.457 − 0.889i)9-s + (0.833 + 0.551i)10-s + (−0.288 − 0.957i)11-s + (0.994 + 0.109i)12-s + (0.322 − 0.946i)13-s + (0.989 + 0.145i)14-s + (−0.957 + 0.288i)15-s + (0.252 + 0.967i)16-s + (−0.424 + 0.905i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1101816569 + 0.1269116589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1101816569 + 0.1269116589i\) |
\(L(1)\) |
\(\approx\) |
\(0.5614265783 - 0.1866159144i\) |
\(L(1)\) |
\(\approx\) |
\(0.5614265783 - 0.1866159144i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (-0.946 - 0.322i)T \) |
| 3 | \( 1 + (0.853 - 0.520i)T \) |
| 5 | \( 1 + (-0.967 - 0.252i)T \) |
| 7 | \( 1 + (-0.983 + 0.181i)T \) |
| 11 | \( 1 + (-0.288 - 0.957i)T \) |
| 13 | \( 1 + (0.322 - 0.946i)T \) |
| 17 | \( 1 + (-0.424 + 0.905i)T \) |
| 19 | \( 1 + (0.145 + 0.989i)T \) |
| 23 | \( 1 + (0.957 + 0.288i)T \) |
| 29 | \( 1 + (-0.976 - 0.217i)T \) |
| 31 | \( 1 + (-0.520 + 0.853i)T \) |
| 37 | \( 1 + (-0.989 + 0.145i)T \) |
| 41 | \( 1 + (0.181 + 0.983i)T \) |
| 43 | \( 1 + (-0.791 + 0.611i)T \) |
| 47 | \( 1 + (-0.581 + 0.813i)T \) |
| 53 | \( 1 + (-0.768 - 0.639i)T \) |
| 59 | \( 1 + (0.357 + 0.934i)T \) |
| 61 | \( 1 + (0.424 + 0.905i)T \) |
| 67 | \( 1 + (-0.520 - 0.853i)T \) |
| 71 | \( 1 + (-0.217 + 0.976i)T \) |
| 73 | \( 1 + (-0.391 - 0.920i)T \) |
| 79 | \( 1 + (-0.813 + 0.581i)T \) |
| 83 | \( 1 + (-0.872 + 0.489i)T \) |
| 89 | \( 1 + (0.997 - 0.0729i)T \) |
| 97 | \( 1 + (0.489 - 0.872i)T \) |
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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
−26.67552407743941167435846295945, −26.23643338253801975754023654162, −25.48272618453564030050010163771, −24.337943657656959964553453047143, −23.26584626434427325031001925341, −22.223172130166107220170225453973, −20.611774612937768176140107892357, −20.10225619627165682856802178316, −19.11995029500969473167554684975, −18.56273445233042259554081234198, −16.95194063096894331335095171917, −15.916108033001731381277728150418, −15.48756983330841547403919472346, −14.46554138169007084441314230119, −13.1029346329055848334371568149, −11.563362917495762414009644207035, −10.55120733040258578136071979443, −9.42538272899206883651350091857, −8.814309196786659774279224922678, −7.32979822933238124599872887186, −6.93899518927528336561296153822, −4.8369108072445358158570848436, −3.453370409054925259742812974495, −2.237834385874044000770485880727, −0.07706540488575360993063621098,
1.29349977948052824459699930419, 3.09497380680603700511546987901, 3.560759606102141536561214465353, 6.10093470685018205115379792372, 7.325282110173533995832217457397, 8.26880001267674324369052189020, 8.89049977622759881882194060650, 10.1794565782425214387375848046, 11.35914801335379903319925224321, 12.64647474481811459339870173171, 13.091528556770399479604956114181, 14.95062905208094039643547323061, 15.78250139100271375828215246622, 16.64540997097518812673718170320, 18.09881484094186185766608110982, 19.05484542191656470187677820733, 19.46047947896915872600503925186, 20.35564110821869299337887221754, 21.29498747830896473841326795873, 22.74898279300409567353420042058, 23.9816960544406189454148751188, 24.87135798308961394750857720882, 25.71822862605672474287471562394, 26.622280168099078043693938047128, 27.29757381649409342483450426918