Yes. Up to normalization one finds, for $\hat e_k$ in a Hermitian basis of the algebra,
$$
\hat C_2=\frac{1}{2}\sum_{k=1}^\ell \hat e_k^2
$$
with eigenvalue
$$
c_2=2\langle \Lambda\vert \delta\rangle +\langle \Lambda\vert\Lambda\rangle
$$
where $\vert\Lambda\rangle=\sum_{i=1}^r \lambda_i\vert w_i\rangle$ is the highest weight state expressed in terms of the fundamental weights $\{\vert w_i\rangle\}$ and $\delta$ is the Weyl root, which is half of the sum of all positive roots.
The details on the derivation of this result can be found in
J. F. Cornwell, Group Theory in Physics (Academic, New York,
1984), Vol. 2.
and worked out examples can be found in
R. Slansky, Phys. Rep. 79, 1 (1981)
or
Iachello, F., 2006. Lie algebras and applications (Vol. 12). Berlin:
Springer