Hash function

Consider a hash function $h$ that computes the hash value of a string in linear time. You are provided with the following description of the hash function:

1. It takes a string as input.

2. It iterates through the string from left to right.

3. In step $i$, it calculates the hash value of the $i$th prefix, i.e. the substring from the first character to the $i$th one. In other words, if the string is $a_1a_2\dots a_n$, at the $i$th step it computes $h(a_1a_2\dots a_i)$.

4. It returns the list of these hash values in the same order they appear concatenated as a string.

Is it a good hash function? If not, which property is missing here?