I deleted some comments, which are think are no longer relevant (an erroneous comment, its correction, and retraction), because of weird typesetting issues that were causing them to spill way off the page, and someone complained. For posterity, here they are:

I cannot reproduce your equality in Mathematica: your rhs is about 49, see code: 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, 30] – Per Alexandersson 2 days ago

@Per I don't have Mathematica. Is it possible you are working with low precision? This Wolfram Alpha query returns error 10^(-15): wolframalpha.com/input/…z_2z_1%2Fz_12^2+-2*z_1^3%2Fz_12^3+-+pi^3+%2F+%284%29%29%2F7%29%2Cz_3%3DRiemannZeta%27%27%27[1%2F2.0]%2Cz_2%3DRiemannZeta%27%27[1%2F2.0]%2Cz_1%3DRiemannZeta%27[1%2F2.0]%2Cz_12%3DAbs%28RiemannZeta%281%2F2.0%29%29 – joro 2 days ago
1

@Per: you are missing a / in the second to last term. – Jack Huizenga 2 days ago

@Jack: Thanks! – Per Alexandersson 2 days ago

Thanks Andres, sorry I forgot to link!