From some known special values of the Riemann zeta function and its derivative, one can show that $$\gamma =1+ \frac{\zeta'(2)}{\zeta(2)} -\frac{\zeta'(0)}{\zeta(0)}+ \frac{\zeta'(-1)}{\zeta(-1)}.$$ Is there a direct proof of this identity, and, if so, does it generalize?
Note also that $$\gamma = \lim_{s\to 1}\left(\frac{\zeta'(s)}{\zeta(s)}+ \frac{1}{s-1} \right)$$
EDIT: I just found the answers here and here:
A kind of reflection formula for the logarithmic derivative of the zeta function
About the logarithmic derivative of the Riemann zeta function
