$s=14 i$
$$\lim_{n \rightarrow \infty}
\left(
\left[
1-
\left(
\int _{-n}^{n} \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta(\tfrac{k}{n}+s)}
\Bigg/
\int _{-n}^{n} \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta(\tfrac{k}{n}+s+\tfrac{1}{n})}
\right)
\right]^{-1}
+\frac1n + s
\right)=0.500000000000000092 + 14.134725141734693381 I$$

$s=14 i$

$$\lim_{n \rightarrow \infty} \left( \left[ 1- \left( \sum _{-n}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta(\tfrac{k}{n}+s)} \Bigg/ \sum _{-n}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta(\tfrac{k}{n}+s+\tfrac{1}{n})} \right) \right]^{-1} +\frac1n + s \right)=0.500000000000000000+14.134725141734693790 i$$

(* start *)
(* Mathematica 8.0.1 *)
n = 100;
s = 0 + 14*I;
s + 1/n +
1/(1 - Sum[(-1)^(k - 1)*Binomial[n - 1, k - 1]/Zeta[s + k/n], {k, 1,
n}]/Sum[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/Zeta[s + k/n + 1/n], {k, 1, n}]);
N[%, n]
(* end *)

(*start*)(*Mathematica 8.0.1*)n = 100;
s = 0 + 14*I;
s + 1/n +
1/(1 - NIntegrate[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/Zeta[s + k/n], {k, -(n*2), (n*2)},
PrecisionGoal -> 20, MaxRecursion -> 20,
WorkingPrecision -> 180]/
NIntegrate[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/
Zeta[s + k/n + 1/n], {k, -(n*2), (n*2)}, PrecisionGoal -> 20,
MaxRecursion -> 20, WorkingPrecision -> 180]);
N[%, 50]
(*end*)

0.499999999999999999999999999999999999999999999890 +
14.134725141734693790457251983562470270784257112386 I