Answer
The answer is 769767316159 / 12574325400 = apx. 61.2173847639572.
Solution
One could use a Markov chain to answer this, but I prefer calculus. The key is that the answer is the same if the time between rolls is the exponentially distributed with a mean of one. That said, the answer can be expressed as the integral from 0 to infinity of:
1-(1-exp(-x/36))^2*(1-exp(-x/18))^2*(1-exp(-x/12))^2*(1-exp(-x/9))^2*(1-exp(-5*x/36))^2*(1-exp(-x/6))
You may easily solve such integrals with an integral calculator.
You may also solve any such problem with my Expected Trials Calculator.