Closed form for $ -\sum_{k=1}^{\infty} \zeta(1-2k)x^{2k} $

Question

Does the following infinite sum converges for real x satisfying $0<x<1$ and if so are there any closed form for it ?

$$ -\sum_{k=1}^{\infty} \zeta(1-2k)x^{2k} $$

Thanks for your help.