In this paper, we prove the equality of the local arithmetic and analytic epsilon- and L-factors attached to the twisted exterior and symmetric square representations of GL(N). We will construct the twisted symmetric square local analytic gamma- and L-factor of GL(N) by applying Langlands-Shahidi method to odd GSpin groups. Then we reduce the problem to the stablity of local coefficients, and eventually prove the analytic stabitliy in this case by some analysis on the asymptotic behavior of certain partial Bessel functions.

Let E/F be a quadratic extension of p-adic fields. The local Langlands correspondence establishes a bijection between n-dimensional Frobenius semisimple representations of the Weil-Deligne group of E and smooth, irreducible representations of GL(n, E). We reinterpret this bijection in the setting of the Weil restriction of scalars Res(GL(n), E/F), and show that the Asai L-function and epsilon factor on the analytic side match up with the expected Artin L-function and epsilon factor on the Galois side.

We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.

<p dir="ltr">In this thesis, we prove the rationality results for the ratio of the critical values of certain <i>L</i>-functions, which appear in the constant term of Eisenstein series associated with the exceptional group <i>G</i>_<em>2</em> over a totally imaginary field. Our methodology builds upon the works of Harder and Raghuram, who established rationality results for special values of Rankin-Selberg <i>L</i>-functions for<i> </i><i>GL</i>_<em>n</em><i>× GL</i>_<em>n'</em> by studying the rank-one Eisenstein cohomology of the ambient group <i>GL</i>_n+n' over a totally real field, as well as its generalization by Raghuram [35] for the case over a totally imaginary field.</p><p dir="ltr">The <i>L</i>-functions in this thesis were constructed using the Langlands-Shahidi method for <i>G</i>_<em>2</em> over a totally imaginary field, attached to maximal parabolic subgroups. This is the first instance of applying the Harder-Raghuram method to an exceptional group, and the first case involving more than one function appearing in the constant term. Our results demonstrate the relationship between the rationality of different <i>L</i>-functions appearing in the constant term, allowing one to prove the rationality of one <i>L</i>-function based on the known rationality result of another <i>L</i>-functions.</p>