  The paper develops the result of second Thomae theorem in hyperelliptic case.
The main formula, called general Thomae formula, provides expressions for
values at zero of the lowest non-vanishing derivatives of theta functions with
singular characteristics of arbitrary multiplicity in terms of branch points
and period matrix. We call these values derivative theta constants. First and
second Thomae formulas follow as particular cases. Some further results are
derived. Matrices of second derivative theta constants (Hessian matrices of
zero-values of theta functions with characteristics of multiplicity two) have
rank three in any genus. Similar result about the structure of order $3$ tensor
of third derivative theta constants is obtained, and a conjecture regarding
higher multiplicities is made. As a byproduct a generalization of Bolza
formulas are deduced.
