  One of the celebrated formulas of Ramanujan is about odd zeta values, which
has been studied by many mathematicians over the years. A notable extension was
given by Grosswald in 1972. Following Ramanujan's idea, we rediscovered a
Ramanujan-type identity for $\zeta(2k+1)$ that was first established by
Malurkar and later by Berndt using different techniques. In the current paper,
we extend the aforementioned identity of Malurkar and Berndt to derive a new
Ramanujan-type identity for $L(2k+1, \chi_1)$, where $\chi_1$ is the principal
character modulo prime $p$. In the process, we encounter a new family of
Ramanujan-type polynomials and we notice that a particular case of these
polynomials has been studied by Lal\'{i}n and Rogers in 2013. Furthermore, we
establish a character analogue of Grosswald's identity and a few more
interesting results inspired from the work of Gun, Murty and Rath.

Number Theory