  We define two types of Witten's zeta functions according to Cartan's
classification of compact symmetric spaces. The type II is the original Witten
zeta function constructed by means of irreducible representations of the simple
compact Lie group U. The type I Witten zeta functions, we introduce here, are
related to the irreducible spherical representations of U. They arise in the
harmonic analysis on compact symmetric spaces of the form U/K, where K is the
maximal subgroup of U. To construct the type I zeta function we calculate the
partition functions of 2d YM theory with broken gauge symmetry using the
Migdal-Witten approach. We prove that for the rank one symmetric spaces the
generating series for the values of the type I functions with integer arguments
can be defined in terms of the generating series of the Riemann zeta-function.
