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C (gcc), 116 bytes

d;g;p;r;f(char*s){for(p=r=1;*s;++s){for(g=0;!g;)for(++p,g=d=2;d<p;++d)g=g&&p%d;r*=pow(p,18160/ *s%80%30%14|1);}d=r;}

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Inputs a string and returns its Gödel number.

Uses the ASCII to Gödel value mapping formula from Arnauld's JavaScript answer.