The Weil conjectures asks for rationality, functional equation, Riemann hypothesis, and degree relating to the Hasse-Weil zeta function of a nonsingular projective variety $X$ over a finite field $k$, which is a product of L-functions. Since you are reading Gelbart, as Artin you can think of these as L-functions associated to the Galois representation of Gal$(\bar k/k)$ acting on each degree of the (etale) cohomology of $X$ respectively.

The Langlands program does not ask for rationality, but to each automorphic representation it does ask for a functional equation and meromorphic continuation for the associated automorphic $L$-function. (This also depends on the representation of the $L$-group.) It is generally believed that these $L$-functions also satisfy the Riemann hypothesis, but this I do not think is explicitly a part of the program.

What you do get is that the functoriality conjecture for $H=\{1\}$ and $G=GL_n$ (p.208 of Gelbart) is what some call Langlands reciprocity: to every Galois representation is associated an automorphic representation, so if you like, you may think of the Hasse-Weil zeta function as (conjecturally) a product of automorphic $L$-functions instead, though I am not sure what one gains from this.

EDIT: Also see the answer to this question. Particularly, Kevin Buzzard's comment that RH given by the Weil conjectures leads to bounds on Satake parameters, i.e., towards the Ramanujan for the corresponding automorphic form.