There is a theorem by Langlands and Shalika ([link](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN356556735_0038%7CLOG_0006&physid=PHYS_0007)) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be uncommon).


Is there an easier way to prove this for the case of cuspidal modular forms?



Non-vanishing of L-function of modular form
Yes, for primitive modular forms (both holomorphic and non-holomorphic) you can in fact something much stronger than the non-vanishing result with much (much!) less machinery than Langlands-Shalika.


It follows from the classic analytic properties of the Rankin-Selberg convolution of L-functions (proved in full in Iwaniec and Kowalski's book on number theory, theorem 5.10 in particular), that under mild assumptions you have a zero-free region which for primitive modular forms comes down to


$$\sigma \geq 1-\frac{c}{\log (\mathfrak{q} (f)(|t|+3))}$$


where $c$ a positive absolute constant.


The proof is standard, and as Peter Humphries mentions in the comments, it directly generalizes the Hadamard-de la Vallée Poussin proof of the zero free region for the Riemann zeta function.


In particular, this obviously implies non-vanishing on $\Re(s)=1$.


