  Conformal blocks in any number of dimensions depend on two variables z, zbar.
Here we study their restrictions to the special "diagonal" kinematics z = zbar,
previously found useful as a starting point for the conformal bootstrap
analysis. We show that conformal blocks on the diagonal satisfy ordinary
differential equations, third-order for spin zero and fourth-order for the
general case. These ODEs determine the blocks uniquely and lead to an efficient
numerical evaluation algorithm. For equal external operator dimensions, we find
closed-form solutions in terms of finite sums of 3F2 functions.
