My Take On Champernowne's Constant

$\large 0.12345678910111213\ldots$

The number above is an irrational decimal number created by concatenating all the positive integers in ascending order, and it is called Champernowne's constant.

Let the $$i^\text{th}$$ digit of fractional part of Champernowne's constant (base 10) be denoted by $$a_{i}$$. Calculate the digital sum of product of all non-zero $$a_{i}$$ where $$i$$ is a prime number less than 1000.

Clarification:

The digital sum is sum of all digits of a number. For example, product of $$a_i$$ where $$i$$ is prime less than $$8$$ is $$2\times 3\times 5\times 7=210$$. Its digital sum is $$2+1+0=3$$.

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