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A355022
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Decimal expansion of Im(Li(3, (i+1)/2)), where Li(n, z) is the polylogarithm function and i is the imaginary unit.
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0
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5, 7, 0, 0, 7, 7, 4, 0, 7, 0, 8, 8, 7, 6, 8, 9, 7, 8, 1, 9, 5, 6, 0, 9, 7, 5, 7, 5, 9, 0, 0, 7, 4, 5, 5, 1, 0, 6, 3, 1, 4, 5, 8, 0, 9, 9, 1, 8, 7, 2, 8, 7, 3, 2, 8, 6, 6, 9, 7, 0, 4, 7, 9, 0, 1, 5, 9, 2, 9, 6, 7, 3, 9, 4, 3, 2, 7, 5, 7, 7, 1, 7, 3, 6, 1, 7, 0, 5, 4, 8, 5, 1, 4, 0, 5, 1, 4, 2, 1, 4, 5, 3, 8, 8, 5
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OFFSET
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0,1
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COMMENTS
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This constant is the subject of the paper by Campbell, Levrie and Nimbran (2021).
The real part of Li(3, (i+1)/2) is 35*zeta(3)/64 - 5*Pi^2*log(2)/192 + log(2)^3/48.
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LINKS
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FORMULA
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Equals (1/16) * Sum_{k>=0} (-1/4)^k/(2*k+1)^3 + (1/2) * Sum_{k>=0} (-1/4)^k/(4*k+1)^3 + (1/4) * Sum_{k>=0} (-1/4)^k/(4*k+3)^3.
Equals (1/2) * Integral_{x=0..1} log(1-x)^2/(1+x^2) dx.
Equals (1/3) * Integral_{x=0..1} arctan(x)*log(1+x)/x dx - G*log(2)/2 + 3*Pi^3/128 + Pi*log(2)^2/32, where G is Catalan's constant (A006752).
Equals Integral_{x=0..1} arctan(x)*log(1-x)/x dx - G*log(2)/2 + 7*Pi^3/128 + Pi*log(2)^2/32.
Equals 23*Pi^3/384 + 3*Pi*log(2)^2/32 - sqrt(2) * Sum_{k>=0} binomial(2*k,k)/(2^(3*k)*(2*k+1)^3) = 23*Pi^3/384 + 3*Pi*log(2)^2/32 - sqrt(2) * 4F3({1/2,1/2,1/2,1/2}, {3/2,3/2,3/2}, 1/2), where 4F3 is the generalized hypergeometric function.
Equals Sum_{k>=1} sin(Pi*k/4)/(2^(k/2)*k^3).
Equals (1/8) * Sum_{k>=0} (-1)^k * H(k)^2/(2*k+1) + 3*Pi^3/128 - 3*Pi*log(2)^2/32, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
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EXAMPLE
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0.57007740708876897819560975759007455106314580991872...
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MATHEMATICA
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RealDigits[Im[PolyLog[3, (I + 1)/2]], 10, 100][[1]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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