My question is how should one think of padic L functions? I know they have been constructed classically by interpolating values of complex Lfunctions. Recently I have seen people think about them in terms of Euler systems. But we know only a few Euler systems and there are lot of padic L functions. In case of elliptic curves(at least over $\mathbb{Q}$) complex Lfunctions give information about the Galois representations. Should the padic Lfunction give some information about some padic Galois representation? It seems to be the case in case of cyclotomic fields where we think of the cyclotomic character as a 1dimensional representation. I apologize in advance if my questions are vague. I am just starting to learn about the subject.

There are three way to obtain $p$adic Lfunctions. The big dream is that one can do all of them for a large class of $p$adic Galois representations $V$. To study them one starts best to look at the cases $\mathbb{Q}(1)$ for the classical KubotaLeopoldt $p$adic $L$functions or the Tatemodule of an elliptic curve etc. Let $K_{\infty}=\mathbb{Q}(\mu_{p^{\infty}})$ be the union of all cyclotomic fields of roots of unity of $p$power order. Let $G$ be its Galois group, which is isomorphic to $\mathbb{Z}_p^{\times}$.
In some sense the Euler system is the bridge between the analytic and the algebraic world. Under the Coleman map it links to the analytic side. In the other direction, one can form derivative classes out of the cohomology classes. These derived classes can be analysed locally and they can be used to bound the Selmer group and hence the characteristic series. That is how one can prove the main conjecture in some cases in one direction. Probably a good place to start is CoatesSujatha. The $p$adic $L$function of an elliptic curve is conjectured to satisfy a $p$adic Birch and SwinnertonDyer formula. (MazurTateTeitelbaum and BernardiPerrinRiou in the supersingular case). On the algebraic side instead, we know almost that the characteristic series satisfies this formula. The order of vanishing is known to be at least as large as the rank and if they agree then the leading term has the desired shape involving the TateShafarevich group; of course only up to a $p$adic unit. In the geometric case, say an elliptic curve over a function field $K$ of a curve over a finite field $k$, the complex and the $p$adic function are the same ($p\neq\text{char}(k)$), since they are both just a polynomial with integer coefficients. Tate's Bourbaki talk on BSD shows how one can use the tower $K_{\infty} = \bar{k} \cdot K$ to prove a good deal about BSD. Iwasawa theory tries to mimic this. So I believe that $p$adic $L$functions are just as nice and interesting as their complex counterparts. Even if they seem more mysterious and the definition is less straight forward, we sometimes know more about them. Now I stop otherwise I am going to write a book about it here... 


It's a sensible question, for this reason: the (classical, complex) Lfunctions are defined in such a way that you can write them down, as Dirichlet series, at least in a right halfplane. What corresponds for padic Lfunctions? Essentially there isn't anything that matches. You can sit in many lectures on padic Lfunctions without seeing anything that merits the name "function", in the sense of function theory. What lies behind this? It is not that padic numbers are less "explicit" than complex numbers at all. It is quite a different reason: the moduletheoretic meaning of a padic Lfunction defines it up to a unit in a certain ring, with the implication that these functions are essentially polynomials. And it is through the modules that Iwasawa theory gains traction in number theory. 

