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For any given integer    k  ⩾  2    we prove the existence of infinitely many   q   and characters    χ    (  mod    q  )    of order   k   such that    |  L   (  1  ,  χ  )   |  ⩾  (   e  γ   +  o   (  1  )   )  log     log  q   . We believe this bound to be the best possible. When the order   k   is even, we obtain similar results for    L  (  1  ,  χ  )    and    L  (  1  ,  χ  ξ  )   , where   χ   is restricted to even (or odd) characters of order   k   and   ξ   is a fixed quadratic character. As an application of these results, we exhibit large even‐order character sums, which are likely to be optimal.

LARGE VALUES OF L ( 1 , χ ) FOR k TH ORDER CHARACTERS χ AND APPLICATIONS TO CHARACTER SUMS