Riemann zeta zeros as zeros of Hermitian polynomial matrix.

Clear[nn, t, n, k, M, x];
nn = 8;
a = N[Table[Total[Im[ZetaZero[Divisors[n]]]], {n, 1, nn}]]
t[n_, 1] = 0;
t[1, k_] = 0;
t[n_, k_] :=
  t[n, k] =
   If[n < k,
    If[And[n > 1, k > 1], x - Sum[t[k - i, n], {i, 1, n - 1}], 0],
    If[And[n > 1, k > 1], x - Sum[t[n - i, k], {i, 1, k - 1}], 0]];
M = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
MatrixForm[M];
FullSimplify[
  Table[x /.
    Solve[Sum[(t[n, k] + a[[GCD[n, k]]])*Cos[-2*Pi*k/n], {k, 1, n}] ==
       0, x], {n, 1, nn}]];
Chop[Sort[Flatten[N[%]]]]
x /. Solve[
  Det[M + Table[Table[a[[GCD[n, k]]], {k, 1, nn}], {n, 1, nn}]] == 0,
  x]