$MinPrecision = 600
$RecursionLimit = ∞

(*These set how much accuracy you want in the infinite sums.*) 
M = 100 

    K = 15

    Gam[s_, x_] := 
      Integrate[
        Exp[-t] (t^(s - 1)), {t, x, ∞}]

    Γ[s_, x_] := Gamma[s, x]

    γ[s_, x_] := 
      Gamma[s] - 
        Sum[(((((-1))^n) (x^(n + s)))/(n!*(n + s))), {n, 
            0, M}]


    Dummy[s_] := 
      y + (2  s - 1) - 
        m - ((s (s - m))/(y + (2  s + 1) - m))

    Gam1[s_] := y + (2*s - 1) - m - ((s (s - m))/Gam1)[s + 1]; 
    Gam1[300] := Dummy[300]

    Gam2[m1_, y1_] := 
      Block[{m = m1, y = y1}, Exp[-y1] y1^m1/Gam1[1]]

    Gam3[m1_, y1_, i_] := Block[{m = m1, y = y1}, Gam1[i]]

    A[s_] := (γ[(s/2), π] + 
            Sum[Gam2[(s/2), π*(((k))^2)]/(k^s), {k, 2, K}])/
        Gamma[(s/2)]

    B[s_] := (π^(s - 1/2))*(Gam[((1 - s)/2), π] + 
              Sum[Gam2[((1 - s)/2), π*(((k))^2)]/(((k))^(1 - s)), {k, 2, K}])/Gamma[(s/2)]

    F[k_] := ((π^(k/2))/(Gamma[k/2]*k (k - 1)))
 (*Z[k] is the zeta calculation.*) 

    Z[k_] := 
      A[k] + B[k] + F[k] // 
        N
 (*This compares our calculation with Mathematica' s built - in zeta function.*) 

    Error[k_] := Z[k] - Zeta[k]