What is the distribution of a sum of constants times Poissons

Question

I have three independent random variables:

$X_i$~$Poisson(i)$, $i=1,2,3$

I have no clue how find the distribution of the sum if I multiply them by some positive constant, say like:

$S= \sum_iiX_i$

for $i=1,2,3$

Without the constants I can just sum up the parameters since they are independent, then S will be:

$S$~$Poisson(6)$

But with the constants out front of each X, I have no idea how to start.