s = -2 n, n element Z, n>=1

s = ρ_n, n!=0, n element Z

-1/2 - 1/2 s log(2 π) + 1/48 s^2 (24 γ_1 + 12 gamma ^2 - π^2 - 12 log^2(2 π)) + 1/48 s^3 (-4 (-3 γ_2 - 6 γ_1 log(2 π) + 2 ζ(3) + log^3(2 π)) + 24 gamma γ_1 + 8 gamma ^3 + 12 gamma ^2 log(2 π) - π^2 log(2 π)) + 1/24 ζ^(4)(0) s^4 + O(s^5) (Taylor series)

2^(-s) + 3^(-s) + 4^(-s) + 5^(-s) + 6^(-s) + 7^(-s) + 8^(-s) + 9^(-s) + 10^(-s) + 11^(-s) + 12^(-s) + 1

lim_(s->∞) ζ(s) = 1

ζ(s) = sum_(k=1)^∞ k^(-s) for Re(s)>1

ζ(s) = ζ(s, 1)

ζ(s) = S_(-1 + s, 1)(1)

ζ(s) = ζ(s, 1/2)/(-1 + 2^s)

ζ(s) = sum_(k=1)^∞ k^(-s) for Re(s)>1

ζ(s) = (2^s sum_(k=0)^∞ (1 + 2 k)^(-s))/(-1 + 2^s) for Re(s)>1

ζ(s) = e^( sum_(k=1)^∞ P(k s)/k) for Re(s)>1

ζ(s) = 1/Γ(s) integral_0^∞ t^(-1 + s)/(-1 + e^t) dt for Re(s)>1

ζ(s) = (-1)^s/(s (-2 + s)!) integral_0^1 (log(1 - t^(-1 + s))/t)^s dt for (s element Z and s>=2)

ζ(s) = 2^(-1 + s)/Γ(1 + s) integral_0^∞ t^s csch^2(t) dt for Re(s)>1

ζ(s) = 2^s π^(-1 + s) Γ(1 - s) sin((π s)/2) ζ(1 - s)

ζ(s) = (π^(-1/2 + s) Γ((1 - s)/2) ζ(1 - s))/Γ(s/2)

ζ(s) = ( sum_(k=0)^∞ (Γ(1 + k - s/2) sum_(j=0)^k (-1)^j (1 + 2 j) binomial(k, j) ζ(2 + 2 j))/(k!))/((-1 + s) Γ(1 - s/2))