Let's breakdown the question in two parts:
1) What is the distribution of my data?
By looking at the plot, we can say that it is a bimodal distribution since it has two peaks (two different modes). (https://en.m.wikipedia.org/wiki/Multimodal_distribution)
2) How to represent it as Standard Gaussian (mean=0 and standard deviation=1)?
Since you have the non-zero mean and a standard deviation, you have already modelled your data as a Gaussian distribution.
From your question, it looks like you want to transform this Gaussian distribution model into a Standard Gaussian distribution model with zero mean and unit standard deviation. The way to do this is to use the Z-value where:
Z = (X - mean) / sd
So, you basically take every point of your data and subtract the mean of 0.0359 and then divide it by the standard deviation of 0.0012. This is will make sure that the new mean of these new points will be zero and the standard deviation will be 1.
