Dirichlet series of Bernoulli polynomials

Question

I am interested in the Dirichlet series $$ \mathcal{B}(k,s) = \sum_{m\geq 1} \frac{B_k(m)}{m^s} $$ where $B_k(x)$ is the $k$th Bernoulli polynomial and $\Re(s) > k + 1$. This converges for all $k$ and $\Re(s) > k + 1$ because the definition $$ B_k(x) = \sum_{i = 0}^k \binom{k}{i}B_ix^{k-i} $$ implies $$ \mathcal{B}(k,s) = \sum_{i=0}^k\binom{k}{i}B_i\zeta(i + s - k). $$ Obviously $$ \lim_{s\rightarrow \infty} \sum_{m \geq 1} \frac{B_k(m)}{m^s} = B_k $$ but are there any other notable properties? In particular, does it have a closed form or interesting bounds? Given the ubiquity of Bernoulli polynomials, I figured the properties of this function would be at least somewhat known, but I have yet to find a single reference on the matter.