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!
