a non principal character in Eq. (56) convergence obtains for Re $s>0$.

Now a suitable definition for generalized Stieltjes constants is (cf. [@lehmer; @knopf]) $$\gamma_k(m)=\lim_{N \to \infty}\left (\sum_{n=2}^N {{\chi(n)} \over {n}}\ln^k n-
{{\ln^{k+1} {N}} \over {m(k+1)}}\right ). \eqno(57)$$

We first note the following from the first equality in (56). We have $$\sum_{n=2}^\infty {{(-1)^n} \over n}(-1)^j L^{(j)}(n,\chi)
=\sum_{k=2}^\infty \chi(k)\ln^j k\sum_{n=2}^\infty {{(-1)^n} \over {nk^n}}$$ $$=\sum_{k=2}^\infty \chi(k)\ln^j k\left[{1 \over k} -\ln\left({{k+1} \over k}\right)
\right]$$ $$=\lim_{x \to \infty}\left \{\sum_{k=2}^x \chi(k){{\ln^j k} \over k}+\sum_{k=2}^x
\chi(k)[\ln k-\ln(k+1)]\ln^j k\right \}.  \eqno(58)$$ Putting $$A(x)=\sum_{k=2}^x \chi(k)[\ln k-\ln(k+1)], \eqno(59)$$ and applying summation by parts we have $$\sum_{k=2}^x \chi(k)[\ln k-\ln(k+1)]\ln^j k=A(x)\ln^j x-j\int_1^x A(t)
{{\ln^{j-1} t} \over t}dt.  \eqno(60)$$

Alternatively, we now proceed in terms of Hurwitz zeta functions and write from the product rule and Eq. (56) $$(-1)^j L^{(j)}(s,\chi)=\sum_{\ell=0}^j (-1)^\ell {j \choose \ell} {{\ln^{j-\ell}m}
\over m^s} \sum_{k=1}^m \chi(k) \zeta^{(\ell)} \left(s,{k \over m}\right).  
\eqno(61)$$ We then have $$\sum_{n=2}^\infty {{(-1)^n} \over n}(-1)^j L^{(j)}(n,\chi)=\sum_{n=2}^\infty {{(-1)^n} \over n}\sum_{\ell=0}^j (-1)^\ell {j \choose \ell} {{\ln^{j-\ell}m}
\over m^n} \sum_{k=1}^m \chi(k) \zeta^{(\ell)} \left(n,{k \over m}\right).$$ $$=\sum_{n=2}^\infty {{(-1)^n} \over n}\sum_{\ell=0}^j {j \choose \ell} {{\ln^{j-\ell}m} \over m^n} \sum_{k=1}^m \chi(k)\sum_{q=0}^\infty {{\ln^\ell(q+k/m)} 
\over {(q+k/m)^n}}$$ $$=\sum_{q=0}^\infty \sum_{\ell=0}^j {j \choose \ell} {{\ln^{j-\ell}m} \over m^n} \sum_{k=1}^m \chi(k)\ln^\ell(q+k/m)\left[{1 \over {mq+k}}-\ln\left({{mq+k+1} \over
{mq+k}}\right)\right].  \eqno(62)$$ The sum on $q$ 