How to evaluate

$$\sum_{n=1}^{\infty}\frac{H_n}{n^52^n}~~,~~\sum_{n=1}^{\infty}\frac{H_n}{n^62^n}$$

In [this post][1] M gives a closed form for the first series,but it contains a zeta function,can it be simplified any more? And is there a closed form for the second series.

[1]: http://math.stackexchange.com/questions ... 2xxdx?lq=1