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Note that Python's official style-guide, PEP8, recommends using lower_case names for variables, so I changed all your Ls to l and all your new_L to new_l.

l = [4, 2, 1, 3]

You should keep track of the cumulative sum in a variable. This way you avoid having to test whether the new_l already has an element in it:

def add_one_by_one(l):
    new_l = []
    cumsum = 0
    for elt in l:
        cumsum += elt
        new_l.append(cumsum)
    return new_l

As an alternative, you could use a generator to avoid having to build the list inside the function (if you are just going to iterate over it later, this is the most memory-efficient way):

def add_one_by_one_gen(l):
    cumsum = 0
    for elt in l:
        cumsum += elt
        yield cumsum

new_l = list(add_one_by_one_gen(l))

# This takes basically no additional memory (only one float/int):
for x in add_one_by_one_gen(l):
    print x

Probably the fastest way to do it would be using the numpy function cumsum:

import numpy as np

new_l = np.cumsum(l)