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In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalizedLambert series, $$  \sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1},  $$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$.Recently, Dixit and the last author pointed out that this series has alreadybeen present in the Lost Notebook of Ramanujan with a more general form.Although, Ramanujan did not provide any transformation identity for it. In thesame paper, Dixit and the last author found an elegant generalization ofRamanujan's celebrated identity for $\zeta(2m+1)$ while extending the resultsof Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored anotherextended version of the aforementioned series, namely,$$\sum_{r=1}^{q}\sum_{n=1}^{\infty}\frac{\chi(r)n^{N-2h}{\exp\left(-\frac{r}{q}n^N x\right)}}{1-\exp({-n^N x})},$$where $\chi$ denotes a Dirichlet character modulo $q$, $N\in 2\mathbb{N}$ andwith some restriction on the variable $h$. In the current paper, we investigatethe above series for {\it any} $N \in \mathbb{N}$ and $h \in \mathbb{Z}$. Weobtain a Dirichlet character analogue of Dixit and the last author's identityand there by derive a two variable generalization of Ramanujan's identity for$\zeta(2m+1)$. Moreover, we establish a new identity for $L(1/3, \chi)$analogous to Ramanujan's famous identity for $\zeta(1/2)$.

A Dirichlet character analogue of Ramanujan's formula for odd zeta values