  This paper is a continuation of our recent papers with the same title,
arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 where a number of integral
equalities involving integrals of the logarithm of the Riemann zeta-function
were introduced and it was shown that some of them are equivalent to the
Riemann hypothesis. A few new equalities of this type, which this time involve
exponential functions, are established, and for the first time we have found
equalities involving the integrals of the logarithm of the Riemann
zeta-function taken exclusively along the real axis. Some of the equalities we
have found are tested numerically. In particular, an integral equality
involving the logarithm of abs(zeta(1/2+it)) and a weight function
cosh(pi*t)^(-1) is shown numerically to be correct up to the 80 digits. For
exponential weight function exp(-at), the possible contribution of the Riemann
function zeroes non-lying on the critical line is rigorously estimated and
shown to be extremely small, in particular, smaller than trillion of digits,
10^(-10^(13)), for a=4*pi.
