L-value calculation

[John Jones]

This is from the feedback page concerning

  https://www.lmfdb.org/L/4/1040/1.1/c1e2/0/0

Hi, I hope this note finds you well. I was wondering how the L(1) approximation on LMFDB was calculated, because I don't believe LMFDB would have calculation errors. At the URL, it says L(1) is 0.42, but I got 0.55. I'm using all the dirichlet coefficients available on LMFDB (approximately 130 terms) and I defined L(s) as the sum of a(n)/n^(-s). I consistently get high percent error for L(1) for various bianchi modular forms, so maybe I'm not using the right formula. I'm sorry for wasting your time if it's me not using the correct/LMFDB definition of the L-function. But if a(n) are the dirichlet coefficients, then I'm confused. Also, why are the L(3/2) values not displayed/available? Wouldn't it be quick to compute those? Thank you. Best, Francis

[Andrew Sutherland]

Magma appears to agree with the LMFDB (and the value in the LMFDB was not computed using Magma, so I'm inclined to believe both are correct):

> R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
> E := EllipticCurve([K![1,1],K![1,1],K![1,0],K![-2,0],K![-1,-1]]);
> L:=LSeries(E);
> Evaluate(L,1);
0.425218322353457645030012715366

My guess is the discrepancy is due to not using enough coefficients (130 is definitely not enough); Magma wants 1418:
> LCfRequired(L);

1418

We generally do not store enough coefficients in the LMFDB to allow L-values to be computed (we do for most of the CMFs but for L-functions coming from motives that can easily be used to compute the coefficients it doesn't make sense to do this, nor would it be feasible beyond a very low conductor bound).

[Francis Bahk]

Hi,

Thank you for the explanations. I had thought 130 coefficients wouldn't be enough but my REU mentor told me otherwise and told me to email, so I'll let him know about this.

Best,

Francis