sidef-scripts/Math/zeta_derivative_at_trivial_zeros.sf

#!/usr/bin/ruby

# The derivatives ζ'(−2n) at trivial zeros.

define π = Num.pi

func zeta_derivative_trivial_zero(n) {
    (-1)**n * zeta(2*n + 1) * ((2*n)!) / 2**(2*n + 1) / π**(2*n)
}

each(1..10, {|n|
    say "zeta'(#{'%3s' % -2*n}) =~ #{zeta_derivative_trivial_zero(n)}"
})

__END__
zeta'( -2) =~ -0.030448457058393270780251530471154776647000483545
zeta'( -4) =~ 0.0079838114502686242806966707987893039052376933623
zeta'( -6) =~ -0.00589975914351593745062987740839202557980153462016
zeta'( -8) =~ 0.00831616198560224735952442651053421422567412291883
zeta'(-10) =~ -0.0189299263381403742289805022903467952319852580952
zeta'(-12) =~ 0.0632705833414630005951823012343077675114181847532
zeta'(-14) =~ -0.291657724743873520321224003070250666970263038533
zeta'(-16) =~ 1.77302566089909639624778734418929448135541982765
zeta'(-18) =~ -13.7427682502140544352205641905185510730953721577
zeta'(-20) =~ 132.28099750421251452709821158578551868064801