$MinPrecision  =  500

M = 
      200
  
p = 5
SP[n_] := 
      NSum[1/(k^(2  n + 1) (Exp[2  π*k] + 1)), {k, 1, M}, 
        NSumTerms -> M]
SM[n_] := 
      NSum[1/(k^(2  n + 1) (Exp[2  π*k] - 1)), {k, 1, M}, 
        NSumTerms -> M]
a = SM[(p - 1)/2]
b = SP[(p - 1)/2]
c = π^p
d = Zeta[p] // N
ZetaMatrix = Round[10^50*{a, b, c, d}]
MatrixForm[
      lattice  =  
        Transpose[
          Append[IdentityMatrix[4],  
            ZetaMatrix]]]
MatrixForm[
      newlattice  =  LatticeReduce[lattice]]
newpoly  =  
      Drop[First[newlattice], -1] . {SM, SP,  PIE,  
          Zeta}
FullSimplify[Solve[newpoly == 0, Zeta]]

3024  a + 84  b - 5  c + 1470  d