While attempting to solve the question Alternating harmonic sum $\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k$ we have discovered a following puzzling identity: \begin{equation} Li_q(-1) = \left(2^{1-q} - 1\right) \zeta(q)=\frac{(-1)^{q-1}}{(q-1)! 2^q} \left( \Psi^{(q-1)}(\frac{1}{2}) - \Psi^{(q-1)}(1)\right) \end{equation} for $q=2,3,4,\cdots$. As far as I know no closed form expressions are known for zeta function values at odd integers. The relationship above does give some closed form expression. Now, the question would be is this relationship something trivial or are there any generalizations of that?
Note: Using the series representation of the polygamma function the above relation simply reduces to: \begin{equation} \Psi^{(q-1)} \left(\frac{1}{2}\right) = \left(2^q-1\right) \Psi^{(q-1)}\left(1\right) =\left(2^q-1\right) (-1)^q (q-1)! \zeta(q) \end{equation} Is this a trivial identity or not ?