Dirichlet generating function with infinitely many zeros with real part 1/2

(* Mathematica start *)
Clear[s, f, zeta];
zeta = Normal[Series[Zeta[s], {s, 1, 0}]]
f[s_] := EulerGamma + 1/(-1 + s)
Reduce[f[s]*f[Conjugate[s]]/f[s + Conjugate[s]] == 0, s]

nn = 100;
s = 7;
c = 8;
N[Sum[Sum[If[GCD[n, k] == 1, 1/(n^s*k^c), 0], {k, 1, nn}], {n, 1,
   nn}], 20]
N[Zeta[s]*Zeta[c]/Zeta[s + c], 20]
(* Mathematica end *)

Output:

EulerGamma + 1/(-1 + s)

s == (-1 + EulerGamma)/EulerGamma || Re[s] == 1/2

1.0124297081184290727

1.0124297081185925265

Maybe relevant:
(*start*)
nn = 200;
s = N[8, 20];
(Sum[Sum[If[Mod[n, k] == 0, (MoebiusMu[k])^(1/2)/n^s, 0], {k, 1,
     nn}], {n, 1, nn}])^2
(A = Table[
    Table[If[Mod[n, k] == 0, (MoebiusMu[k])^(1/2)/n^s, 0], {k, 1,
      nn}], {n, 1, nn}]);
N[Total[Total[A.Transpose[A]]], 20]
Zeta[s]*Zeta[s]/Zeta[2*s]
(*end*)

Output:

1.00815593032900188265 + 0.008189189928767649031 I

1.0081559303290019133

1.008155930329001935