This might be naive question but I was wondering whether a p-adic analogue of the following (shockingly) beautiful formula $$\zeta(s)\Gamma(s) = \int_0^\infty \frac{t^{s-1}}{e^t-1} dt$$ (vaild for $\mathrm{Re}(s)> 1$ and with all the usual notations) of Riemann is (well) known.

So with a p-adic analogue I'm of course aiming at an integral expression for the product of the classical p-adic zeta function $\zeta_p$ that interpolates $\zeta(s)$ in its values at the negative even integers and $\Gamma_p$ might be taken to be Morita's p-adic Gamma function. (Note that up to a power of $t$ the right hand side of the above formula can be said (in a catchy way) to be the Mellin transform of the generating function of the Bernoulli numbers (which essentially give the values of $\zeta(s)$ at the negative even integers...)).

What is "of course" lurking/hidden behind the above formula is the functional equation of the (completed) Riemann zeta function (for the time being I haven't learned/understood yet how this works in the p-adic setting, but I heard that in the work of Perrin-Riou, e.g., questions concerning functional equations of zeta functions in the p-adic world are dealt with (at least conjecturally)).    

So, in a broader sense my question aims at understanding how the functional equation works in the p-adic setting and whether there are "nice functions" behind, that implement the mechanism (in the complex, global setting this is related, e.g., to theta functions and Bernoulli numbers, and the above formula is one incarnation of the eternal beauty of this area of the (mathematical) universe). 

Thank you very much in advance for any help!