Towards lacunar zero free regions.

(*start*)Clear[s, nn, n, k];
nn = 500;
s = 1;
a = %;
lower = N[
  s - Sum[Sum[(-1)^n*
      Binomial[n - 1,
       k - 1]*(-((2^(-k + n) 3^(-1 + k))^-s/
          Log[2^(-k + n) 3^(-1 + k)])), {k, 1, n}], {n, 2, nn}]]
s = 2;
b = %;
upper = N[
  s - Sum[Sum[(-1)^n*
      Binomial[n - 1,
       k - 1]*(-((2^(-k + n) 3^(-1 + k))^-s/
          Log[2^(-k + n) 3^(-1 + k)])), {k, 1, n}], {n, 2, nn}]]

upper - lower

NIntegrate[1/(1 + 1/2^s + 1/3^s), {s, a, b}]
(*end*)

(* Mick *)
https://math.stackexchange.com/q/612879/8530