  Recently, it was conjectured that the first generalized Stieltjes constant at
rational argument may be always expressed by means of Euler's constant, the
first Stieltjes constant, the $\Gamma$-function at rational argument(s) and
some relatively simple, perhaps even elementary, function. This conjecture was
based on the evaluation of $\gamma_1(1/2)$, $\gamma_1(1/3)$, $\gamma_1(2/3)$,
$\gamma_1(1/4)$, $\gamma_1(3/4)$, $\gamma_1(1/6)$, $\gamma_1(5/6)$, which could
be expressed in this way. This article completes this previous study and
provides an elegant theorem which allows to evaluate the first generalized
Stieltjes constant at any rational argument. Several related summation
formul{\ae} involving the first generalized Stieltjes constant and the Digamma
function are also presented. In passing, an interesting integral representation
for the logarithm of the $\Gamma$-function at rational argument is also
obtained. Finally, it is shown that similar theorems may be derived for higher
Stieltjes constants as well; in particular, for the second Stieltjes constant
the theorem is provided in an explicit form.
