L(s) = 1 | + (0.663 + 0.748i)2-s + (−0.239 + 0.970i)3-s + (−0.120 + 0.992i)4-s + (−0.464 + 0.885i)5-s + (−0.885 + 0.464i)6-s + (0.748 − 0.663i)7-s + (−0.822 + 0.568i)8-s + (−0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (0.354 − 0.935i)11-s + (−0.935 − 0.354i)12-s + (0.120 + 0.992i)13-s + (0.992 + 0.120i)14-s + (−0.748 − 0.663i)15-s + (−0.970 − 0.239i)16-s + (−0.568 + 0.822i)17-s + ⋯ |
L(s) = 1 | + (0.663 + 0.748i)2-s + (−0.239 + 0.970i)3-s + (−0.120 + 0.992i)4-s + (−0.464 + 0.885i)5-s + (−0.885 + 0.464i)6-s + (0.748 − 0.663i)7-s + (−0.822 + 0.568i)8-s + (−0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (0.354 − 0.935i)11-s + (−0.935 − 0.354i)12-s + (0.120 + 0.992i)13-s + (0.992 + 0.120i)14-s + (−0.748 − 0.663i)15-s + (−0.970 − 0.239i)16-s + (−0.568 + 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08398917263 + 1.719204211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08398917263 + 1.719204211i\) |
\(L(1)\) |
\(\approx\) |
\(0.7636303680 + 1.056330321i\) |
\(L(1)\) |
\(\approx\) |
\(0.7636303680 + 1.056330321i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (0.663 + 0.748i)T \) |
| 3 | \( 1 + (-0.239 + 0.970i)T \) |
| 5 | \( 1 + (-0.464 + 0.885i)T \) |
| 7 | \( 1 + (0.748 - 0.663i)T \) |
| 11 | \( 1 + (0.354 - 0.935i)T \) |
| 13 | \( 1 + (0.120 + 0.992i)T \) |
| 17 | \( 1 + (-0.568 + 0.822i)T \) |
| 19 | \( 1 + (0.992 - 0.120i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 + (0.354 + 0.935i)T \) |
| 31 | \( 1 + (-0.935 + 0.354i)T \) |
| 37 | \( 1 + (0.970 + 0.239i)T \) |
| 41 | \( 1 + (0.935 + 0.354i)T \) |
| 43 | \( 1 + (0.970 - 0.239i)T \) |
| 47 | \( 1 + (0.885 - 0.464i)T \) |
| 59 | \( 1 + (-0.885 + 0.464i)T \) |
| 61 | \( 1 + (-0.822 + 0.568i)T \) |
| 67 | \( 1 + (0.992 + 0.120i)T \) |
| 71 | \( 1 + (-0.239 - 0.970i)T \) |
| 73 | \( 1 + (-0.822 - 0.568i)T \) |
| 79 | \( 1 + (-0.663 + 0.748i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.568 - 0.822i)T \) |
| 97 | \( 1 + (0.885 + 0.464i)T \) |
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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
−32.20157594113143458818048563920, −30.975463085918037768877784444784, −30.61729233978230793635301284780, −29.13009379400584448444971455534, −28.258689914638458338301721714797, −27.45164865526024728068947249748, −24.86544161525855734010834848042, −24.53583600587501213041166262272, −23.20939821448646245158118130118, −22.34408707274663145520569208319, −20.57415730380003167081245176208, −20.014140288672390067810234299341, −18.55467250291734211393852095592, −17.59757835975270394714567681303, −15.66555050985921680554894817643, −14.31519703678846779225643697951, −12.89632711129503542316847951576, −12.15933857647147128966674205349, −11.21075126475371884841185099858, −9.18530796347847624574783539607, −7.66762920784399821664277060344, −5.74698939540895657198993274005, −4.61053333125827324229436883048, −2.38760518192168568018038571410, −0.864610723824046613682768647261,
3.40635317811632050353762480088, 4.38053539485920063793822484232, 5.98254754197649779039224544427, 7.40151676514073096139655319941, 8.91179307287261369257329822394, 10.90862663976469880714560503282, 11.64742895634198169823647467495, 13.89171184895624092707195909783, 14.5966197405253935896467027831, 15.79501175256772159134134108439, 16.7904996125700885580298859263, 18.00020672654085707194557265746, 19.90927513668940826388676654425, 21.45223148653821591938137100452, 22.02129775348037174107505974606, 23.33497747957582465271525221518, 24.08234047955186716642472486932, 25.91527964570394885713632437166, 26.72493440534073150926923536884, 27.373711523720840562324410339007, 29.29244299298191526091003186809, 30.63108818806059525058006305419, 31.41695072182452242727861219832, 32.82217337885696460462020983877, 33.491245273368252014802179011856