This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.

My understanding is that nowadays there are conjectures which essentially predict (perhaps up to a sign) the value $L(M,n)$, where $M$ is a motive, $L$ is its $L$-function, and $n$ is an integer. My understanding of the history is that (excluding classical works on rank 1 motives from before the war) Deligne realised how to unify known results about $L$-functions of number fields and the B-SD conjecture, in his Corvallis paper, where he made predictions of $L(M,n)$, but only up to a rational number and only for $n$ critical. Then Beilinson extended these conjectures to predict $L(M,n)$ (or perhaps its leading term if there is a zero or pole) up to a rational number, and then Bloch and Kato went on to nail the rational number.

Nowadays though, many motives have $p$-adic $L$-functions (the toy examples being number fields and elliptic curves over $\mathbf{Q}$, perhaps the very examples that inspired Deligne), and these $p$-adic $L$-functions typically interpolate classical $L$-functions at critical values, but the relationship between the $p$-adic and classical $L$-function is (in my mind) a lot more tenuous away from these points (although I think I have seen some formulae for $p$-adic zeta functions at $s=0$ and $s=1$ that look similar to classical formulae related arithmetic invariants of the number field).

So of course, my question is: is there a conjecture predicting the value of $L_p(M,n)$, for $n$ an integer, and $L_p$ the $p$-adic $L$-function of a motive? Of course that question barely makes sense, so here's a more concrete one: can one make a conjecture saying what $\zeta_p(n)$ should be (perhaps up to an element of $\mathbf{Q}^\times$) for an integer $n\geq2$ and $\zeta_p(s)$ the $p$-adic $\zeta$-function? My understanding of Iwasawa theory is that it would only really tell you information about the places where $\zeta_p(s)$ vanishes, and not about actual values---Iwasawa theory is typically only concerned with the ideal generated by the function (as far as I know). Also, as far as I know, $p$-adic $L$-functions are not expected to have functional equations, so the fact that we understand $\zeta_p(s)$ for $s$ a negative integer does not, as far as I know, tell us anything about its values at positive integers.