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

General derivative Thomae formula for singular half-periods