I am interested in the function $$T(s)=\sum_{n=1}^\infty \frac{1}{ ^{s}n}$$ where $s$ is an integer and $ ^s n$ is $n^{n^{n^{...}}}$ $s$ times. I know these series are convergent since tetration of $n>1$ results in incredibly large values. I am also familiar with the Sophomore's Dream Identity $$\int_{0}^1 x^{-x}dx=\sum_{n=1}^\infty \frac{1}{n^n}=T(2)$$ and $T(1)$ is the harmonic series.
My question is does $T(s)$ have an integral representation for other values of $s$? And if not, are there other identities related to this function?