$MinPrecision = 
      50
  
L[s_] := 
      NSum[(3  n + 1)^-s - (3  n + 2)^-s, {n, 0, 100000}, 
        NSumTerms -> 100000]

(*An approximate formula .   This could be improved by using the Gam2 function from the zeta function program but it wasn' t necessary in our case,  
      as we did not require as many digits of accuracy.*) 
M = 30

    K = 20

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

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

    G[m_, 
        y_] := ((Exp[-y] y^m)/(y + 1 - m - (((1 - m))/(y + 3 - m - ((2 (2 - m))/(y + 5 - m - ((3 (3 - m))/(y + 7 - m - ((4 (4 - m))/(y + 9 - m - ((5 (5 - m))/(y + 11 - m - ((6 (6 - m))/(y + 13 - m - ((7 (7 - m))/(y + 15 - m))))))))))))))))

    A[s_] := 
      NSum[γ[((s + 1)/2), ((π*(((3  k + 1))^2))/3)]/(((3  k + 1))^s) - γ[((s + 1)/2), ((π*(((3  k + 2))^2))/3)]/(((3  k + 2))^s), {k, 0, 0}]/Gamma[((s + 1)/2)] + 
        NSum[G[((s + 1)/2), ((π*(((3  k + 1))^2))/3)]/(((3  k + 1))^s) - 
              G[((s + 1)/2), ((π*(((3  k + 2))^2))/3)]/(((3  k + 2))^s), {k, 1, K}, NSumTerms -> K]/Gamma[((s + 1)/2)]

    B[s_] := (Sqrt[3])*(π^(s - 1/2))*(3^(-s))*(NSum[Γ[((2 - s)/2), ((π*(((3  k + 1))^2))/3)]/(((3  k + 1))^(1 - s)) - Γ[((2 - s)/2), ((π*(((3  k + 2))^2))/3)]/(((3  k + 2))^(1 - s)), {k, 0, 0}]/Gamma[((s + 1)/2)] + 
            NSum[G[((2 - s)/2), ((π*(((3  k + 1))^2))/3)]/(((3  k + 1))^(1 - s)) - 
                  G[((2 - s)/2), ((π*(((3  k + 2))^2))/3)]/(((3  k + 2))^(1 - s)), {k, 1, K}, NSumTerms -> K]/
              Gamma[((s + 1)/2)])

    Error[s_] := A[s] + B[s] - L[s]

Error[2]

Plot[{A[n] + B[n] - L[n]}, {n, 5, 20}]