Coefficients of the L-function
Two additional functions related to the global L-function are available which do not require its full
initialization: hgmcoefs(H,t,n) computes the first n coefficients of the L-function by setting
all wild Euler factors to 1; this is identical to lfunan(L,n) when this hypothesis is correct (in
particular if there are no wild primes) otherwise all coefficients divisible by a wild prime will be
wrong. In the above example, only 5 is wild and L5 is indeed trivial, so all is fine.
The second is the function hgmcoef(H,t,n) which only computes the nth coefficient of the
global L-function. It gives an error if n is divisible by a wild prime.
? hgmcoefs(H, 1/64, 7^6)[7^6] \\ slow: 7^6 > 10^6
time = 1min, 1,564 ms.
%26 = -25290600
? hgmcoef(H, 1/64, 7^6)
%27 = -25290600
? hgmcoef(H, 1/64, 10)
Atelier 2022 (13/01/2022) – p. 12/12