Riemann–von Mangoldt formula: Is $\ln(T/2\pi!)$ the number of non-trivial zeroes along the critical line of the zeta function?

Question

From the Riemann–von Mangoldt formula article in Wikipedia:

The formula states that the number N(T) of zeros of the zeta function with imaginary part greater than 0 and less than or equal to T satisfies

$$ N(T)=\frac{T}{2\pi}\ln{\frac{T}{2\pi}}-\frac{T}{2\pi }+O(\ln{T}). $$

But this just looks like an example of Stirling's approximation, so is a better expression simply

$$ N(T)\sim\ln\left(\frac{T}{2\pi}!\right)=\ln\Gamma\left(\frac{T}{2\pi}+1\right) $$ or do the higher-order terms differ?