List of first n prime numbers most efficiently and in shortest code
Python (129 chars, O(n*log log n), score of 203.948)

I'd say the Sieve of Eratosthenes is the way to go. Very simple and relatively fast.

N=15485864
a=[1]*N
x=xrange
for i in x(2,3936):
 if a[i]:
  for j in x(i*i,N,i):a[j]=0
print [i for i in x(len(a))if a[i]==1][2:]


Improved code from before.

Python (191 156 152 chars, O(n*log log n)(?), score of 252.620(?))

I cannot at all calculate complexity, this is the best approximation I can give.

from math import log as l
n=input()
N=n*int(l(n)+l(l(n)))
a=range(2,N)
for i in range(int(n**.5)+1):
 a=filter(lambda x:x%a[i] or x==a[i],a)
print a[:n]


n*int(l(n)+l(l(n))) is the top boundary of the nth prime number.