I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function.

Could the constant term of $\mathrm{GL}(2)$-Eisenstein series be used to prove the functional equation of $L$-functions? (standard on $\mathrm{GL}(2)$ or Rankin-Selberg on $\mathrm{GL}(2) \times \mathrm{GL}(2)$ or adjoint/symmetric square on $\mathrm{GL}(2)$)