Dirichlet eta determinant1 - determinant2

(*start*)
Clear[m, n, k, s, determinant1, determinant2]
m = 7; (*m is the integer variable to vary*)
determinant1 =
 FullSimplify[
  Det[A = (Table[
       Table[(-I)^(k + n)/(k^(s/2)*n^(s/2)), {k, 1, m}], {n, 1, m}] -
      IdentityMatrix[m])]]
TableForm[A]
A[[1, 1]] = 0;
B = A;
TableForm[B]
determinant2 = Expand[FullSimplify[Det[B]]]
"here"
Expand[(-1)^(m)*(determinant1 - determinant2)/2]
Sum[(-1)^(n + 1)/n^s, {n, 1, m}]
FullSimplify[%% - %]
Expand[%]
(*end*)

https://math.stackexchange.com/q/673934/8530
https://math.stackexchange.com/q/4222890/8530