Riemann Zeta Function Calculator
Calculate the Riemann zeta function ζ(s) for real arguments.
Function Parameters
Zeta Function Definition
ζ(s) = Σ 1/n^s = 1 + 1/2^s + 1/3^s + ...
Converges for Re(s) > 1
Quick Values
Riemann Zeta ζ(2)
1.6449350667
Derivative ζ'(2)
-0.93755517
Euler Product (15 primes)
1.63856796
Error: 6.3671e-3
Special Values
ζ(2) = π²/61.64493407
ζ(4) = π⁴/901.08232323
ζ(6) = π⁶/9451.01734306
ζ(3) - Apéry1.20205690
ζ(-1) = -1/12-0.08333333
ζ(0) = -1/2-0.50000000
Integer Values
ζ(2)1.64493507
ζ(3)1.20205690
ζ(4)1.08232323
ζ(5)1.03692776
ζ(6)1.01734306
ζ(7)1.00834928
ζ(8)1.00407736
ζ(9)1.00200839
ζ(10)1.00099458
Properties
- • Pole at s = 1
- • Trivial zeros at s = -2, -4, -6, ...
- • ζ(s) = Π_p (1-p^(-s))^(-1) (Euler)
- • Related to prime distribution