<s>This is just partial numerics, but the following Mathematica code 
strengthens the conjecture, with 500 decimal places:</s>

Here is code for those that want to perform numerics in Mathematica:

    prec = 500; (* Precision of calculations. *)
    lhs = N[Zeta[3], prec]
    rhs = N[(-Zeta'''[1/2]/Abs[Zeta[1/2]] -
     3 Zeta''[1/2] Zeta'[1/2]/Abs[Zeta[1/2]]^2 -
     2 Zeta'[1/2]^3/Abs[Zeta[1/2]]^3 - Pi^3/4)/7, prec]
    lhs == rhs (* Gives true *)