Riemann-Siegel function Z: Series representations (formula 10.04.06.0011)

RiemannSiegelZ

$Mathematica Notation$

Zeta Functions and Polylogarithms

RiemannSiegelZ[z]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/10.04.06.0011.01

Input Form

RiemannSiegelZ[z] \[Proportional] (Exp[-((I (4 z^2 + Pi Sqrt[z^2]))/(8 z))] (z^2)^((I z)/4) (1 + (3 I)/(16 z) - 9/(512 z^2) + (183 I)/(8192 z^3) - 2277/(524288 z^4) + (212829 I)/(8388608 z^5) - 1364445/(268435456 z^6) + (326341455 I)/(4294967296 z^7) - 8198081325/(549755813888 z^8) + (3781776345585 I)/(8796093022208 z^9) - 23339010744567/ (281474976710656 z^10) + (17654423117199729 I)/ (4503599627370496 z^11) - 215619469740469809/(288230376151711744 z^12) + (241858525676475612513 I)/(4611686018427387904 z^13) - 1468114834103562061701/(147573952589676412928 z^14) + (2284179415871077852696767 I)/(2361183241434822606848 z^15) + O[1/z^16]) Zeta[I z + 1/2])/(4^((I z)/4) Pi^((I z)/2)) /; Abs[Arg[z^2]] < Pi && (Abs[z] -> Infinity)

Standard Form

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MathML Form

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<power /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 512 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 183 </cn> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 8192 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2277 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 524288 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn 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Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29