  The connection between derivatives of $L(s,f)$ for periodic arithmetical
functions $f$ at $s=1$ and generalized Stieltjes constants has been noted
earlier. In this paper, we utilize this link to throw light on the arithmetic
nature of $L'(1,f)$ and certain Stieltjes constants. In particular, if $p$ is
an odd prime greater than $7$, then we deduce the transcendence of at least
$(p-7)/2$ of the generalized Stieltjes constants, $\{ \gamma_1(a,p) : 1 \leq a
< p \}$, conditional on a conjecture of S. Gun, M. R. Murty and P. Rath.
