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The principal aim of this article is to attach and study $p$-adic$L$-functions to cohomological cuspidal automorphic representations $\Pi$ of$\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. Weuse a modular symbol approach, along the global lines of the work of Ash andGinzburg, but our results are more definitive since we draw heavily upon themethods used in the recent and separate works of all the three authors. Byconstruction our $p$-adic $L$-functions are distributions on the Galois groupof the maximal abelian extension of $F$ unramified outside $p\infty$. Moreoverwe work under a weaker Panchishkine type condition on $\Pi_p$ rather than thefull ordinariness condition. Finally, we prove the so-called Manin relationsbetween the $p$-adic $L$-functions at all critical points. This has thestriking consequence that, given a unitary $\Pi$ whose standard $L$-functionadmits at least two critical points, and given a prime $p$ such that $\Pi_p$ isordinary, the central critical value $L(\tfrac12, \Pi\otimes\chi)$ is non-zerofor all except finitely many Dirichlet characters $\chi$ of $p$-powerconductor.

$L$-functions of ${\mathrm{GL}}(2n):$ $p$-adic properties and non-vanishing of twists