  We establish computational results concerning the Lagrangian capacity,
originally defined by Cieliebak-Mohnke. More precisely, we show that the
Lagrangian capacity of a 4-dimensional convex toric domain is equal to its
diagonal. Working under the assumption that there is a suitable virtual
perturbation scheme which defines the curve counts of linearized contact
homology, we extend the previous result to any convex or concave toric domain.
This result gives a positive answer to a conjecture of Cieliebak-Mohnke for the
Lagrangian capacity of the ellipsoid.
