  We examine an unstudied manuscript of N.~S.~Koshliakov over $150$ pages long
and containing the theory of two interesting generalizations $\zeta_p(s)$ and
$\eta_p(s)$ of the Riemann zeta function $\zeta(s)$, which we call
\emph{Koshliakov zeta functions}. His theory has its genesis in a problem in
the analytical theory of heat distribution which was analyzed by him. In this
paper, we further build upon his theory and obtain two new modular relations in
the setting of Koshliakov zeta functions, each of which gives an infinite
family of identities, one for each $p\in\mathbb{R^{+}}$. The first one is a
generalization of Ramanujan's famous formula for $\zeta(2m+1)$ and the second
is an elegant extension of a modular relation on page $220$ of Ramanujan's Lost
Notebook. Several interesting corollaries and applications of these modular
relations are obtained including a new representation for $\zeta(4m+3)$.
