By starting with some Dirichlet series similar to logarithms, or somewhat similar to logarithm Dirichlet series, I arrived at this expression:
$\left(\text{ArcSin}\left[\text{Sqrt}[2]*\text{Cos}[\text{Im}[\text{ZetaZero}[2]]*\text{Log}[2]]\left/\left(\frac{1}{18} \left(\sqrt{3} \pi -\sqrt{6} \pi \text{Cos}[\text{Im}[\text{ZetaZero}[1]] \text{Log}[2]]+9 \text{Log}[3]+9 \sqrt{2} \text{Cos}[\text{Im}[\text{ZetaZero}[1]] \text{Log}[2]] \text{Log}[3]\right)\right)\right.\right]*2\right)= -\pi$
Can it be simplified and is the real part the left hand side equal to $-\pi$ ?
I apologize for the bad Latex formatting.
As a Mathematica command the the real part of the left hand side is:
N[Re[(ArcSin[ Sqrt[2]*Cos[ Im[ZetaZero[2]]*Log[2]]/(1/ 18 (Sqrt[3] \[Pi] - Sqrt[6] \[Pi] Cos[Im[ZetaZero[1]] Log[2]] + 9 Log[3] + 9 Sqrt[2] Cos[Im[ZetaZero[1]] Log[2]] Log[3]))]*2)], 90]
= -3.1415926535897932384626433832795028841971693993751058209749445923078\ 1640628620899862803483
Edit 31.8.2012:
This is what the Latex should look like:
= -3.1415926535897932384626433832795028841971693993751058209749445923078\ 1640628620899862803483