  The purpose of this article is twofold. First, we introduce the constants
$\zeta_k(\alpha,r,q)$ where $\alpha \in (0,1)$ and study them along the lines
of work done on Euler constant in arithmetic progression $\gamma(r,q)$ by
Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants
are used for evaluation of certain integrals involving error term for Dirichlet
divisor problem with congruence conditions and also to provide a closed form
expression for the value of a class of Dirichlet L-series at any real critical
point. In the second half of this paper, we consider the behaviour of the
Laurent Stieltjes constants $\gamma_k(\chi)$ for a principal character $\chi.$
In particular, we study a generalization of the "Generalized Euler constants"
introduced by Diamond and Ford in 2008. We conclude with a short proof for a
closed form expression for the first generalized Stieltjes constant
$\gamma_1(r/q)$ which was given by Blagouchine in 2015.
