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

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 *)

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

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 *)