I'm wondering if there is any information on the inverse of the Riemann zeta function (not it's reciprocal, but its functional inverse). This would obviously be a multivalued function.

I fiddled with this recently; however I did not yet arrive at an interesting result for the inverse of the $\zeta$. But one could use the alternatingzeta (or Dirichlet's eta $\eta$) function. It is not too difficult to construct an invertible powerseries for the related etafunction; one can simply consider the sequence of formal powerseries for $ {1 \over (1+1)^x }, {1 \over (1+2)^x}, ... $ and adds the coefficients at the same powers of x. These produce nonconvergent series, but which can be Eulersummed. In Pari/GP one does simply:
and gets $ \small pseta(x)= 0.500000
+ 0.225791 x
 0.0305146 x^2
 0.00391245 x^3
+ 0.00208483 x^4
 0.000312274 x^5 + O(x^{6})
$
$ \small \begin{array} {lll} pszeta(x) &=&  0.500000 + (0.08106151) x  (0.00317823+1) x^2  (0.000785194+1) x^3 \\\ & & + (0.0001207001) x^4  (0.00000194090+1) x^5 + O(x^{6}) \end{array} $ (That powerseries is related to the power series using the Stieltjesconstant by replacing x by x+1 ) The power series for $\eta$ can be recentered at the fixpoint $ \small fp \sim 0.629334 $ to get a powerseries without a constant term which can then be inverted. Let's call this $\eta_{fp} = \eta (x+fp)fp $ then the powerseries begins like $ \small \begin{array} {lll} pseta_{fp}(x) &=& 0.184574 x  0.0337023 x^2 + 0.000152965 x^3 + 0.00117594 x^4 \\\ & &  0.000254950 x^5 + 0.0000216757 x^6 + 0.00000147274 x^7  0.000000714222 x^8 + O(x^{9}) \end{array} $ From this we can generate a powerseries for the inverse of $ \eta_{fp}$. The range of convergence is small; but using eulersummation one can compute values for the inverse of $ \eta_{fp}$. Even fractional iterates are accessible; here is a plot which shows the fractional iteration of $\eta(x,h)$, beginning at x=1 where h is the iterationparameter (all is computed using the centered version $\eta_{fp}$ ). The plot has to be read that at h=0 we have $\eta(x,0)=x=1 $, at h=1 we have $\eta(x,1)=\eta(x)= \log(2)$, at h=2 we have $\eta(x,2)=\eta(\eta(x))$ at h>inf we get the fixpoint fp and the inverse is at h=1: $ \eta(x,1)=\eta^{1}(x) \to \infty $
I'm not yet ready with a small script/sketch of an article where I explore this in a bit more detail. [update] Here is a plot for a range of the inverse alternating zeta; for the "extreme" values at the borders I used Eulersummation because the power series has very small range of convergnce. 


The question is about the value distribution of $\zeta(s)$; it is considered (without speaking of inverse) in some detail in Chapter XI of Titchmarsh's book. 


The following Mathematica program gives the first few zeta zeros to good accuracy by applying series reversion or the inverse as you call it, twice:
14.1347, 21.022, 25.0109, 30.4249, 32.9351, 37.5862, 40.9187, 43.3271, 48.0052, 49.7738, 52.9703, 56.4462, 59.347, 60.8318, 65.1125, 67.0798, 69.5464, 72.0672, 75.7047, 77.1448 Of course by giving not the approximation of the zeros as a starting point, but the zeros them selves it converges even faster for $x=0$. Expanding for other values not close to zeros the function appears to be much less orderly when looking at a few values. 

