Update: @arijit: to answer the question you asked in the comments of David's answer, I think what you do is the following: In the ordinary case, the Selmer group one generally looks at is the one Greenberg defined, i.e. the "ordinary" Selmer group whose definition is based on the existence of the subrepresentation on which Frobenius acts via the unit root $\alpha$. I.e. there is a sub $W\subseteq V$ and the local condition of the Selmer group is
$$\ker\left(H^1(D_p,V)\rightarrow H^2(I_p,V/W)\right)$$
Accordingly, you should be looking at the $p$-adic $L$-function given by $\alpha$. Now, the filtered $(\varphi,N)$-module $D=D_{\text{cris}}(V)$, has two $\varphi$-stable subspaces: the one coming from the bona fide subrepresentation, namely $D_{\text{cris}}(W)$, and a non-admissible sub $D^\prime$ coming from the non-unit root. If you use this sub, you can define a local condition for a Selmer group completely analogously to the Greenberg defintion, but in the cohomology of $(\phi,\Gamma)$-modules. This should be related to the critical $p$-adic $L$-function. Basically, different "refinements" of the filtered $(\varphi,N)$-module (or "triangulations" of the $(\phi,\Gamma)$-module, or "$p$-stabilizations" of the automorphic representation) should correspond to different Selmer groups and different $p$-adic $L$-functions. Someone please correct me if I've simply made this up!
The interpolation factor is the determinant of $1-\varphi$ on $D_{\text{cris}}$ divided by its determinant on $D_{\text{cris}}^\ast(1)$. As to why this is what it should be, you can trace that back to Coates & Perrin-Riou's original paper (p-adic L-functions attached to motives over Q) where right above their definition of the interpolation factor (equation 4.11) they say "Following a suggestion of R. Greenberg". Deligne suggested an interpretation of the interpolation factor as modifying some $\varepsilon$-factors in a way completely analogous to the modifications of the Gamma-factors at $\infty$, this appears in the papers Coates wrote after the Coates–Perrin-Riou paper, of which Motivic p-adic L-functions in the Durham proceedings is the most definitive account. So that's a couple of ways of thinking about the Euler factors that appear in the interpolation property.