A Simple L-Function

Perhaps the simplest L-function at the LMFDB site (besides the Riemann zeta function) is,

L(χ, s) where χ is the Dirichlet character with label 3, 2, which is given as

1 – 1/2^s + 1/4^s – 1/5^s + 1/7^s – 1/8^s + 1/10^s – 1/11^s + …,

with a product over the primes expression as

1/(1 + 1/2^s) * 1/(1 + 1/5^s) * 1/(1 – 1/7^s) * 1/(1 + 1/11^s) * …

So when on dividing each successive number n by 3, if remainder is 0, then χ(n) = 0; if remainder is 1, χ(n) = 1; if remainder is 2, χ(n) = – 1.  

When s is a positive odd integer the above L-function, results in a simple expression of the form kπ^s/√3.

For example when s = 1,

1 – 1/2 + 1/4 – 1/5 + 1/7 – 1/8 + 1/10 – 1/11 + …  = 2/3 * 5/6 * 7/6 * 11/12 * … 

= π/(3√3).

And when s = 3,

1 – 1/2³ + 1/4³ – 1/5³ + 1/7³ – 1/8³ + 1/10³ – 1/11³ + …

= 8/9 * 125/126 * 343/342 * 1331/1332 * …

= 4π³/(81√3).

And when s = 5,

1 – 1/2⁵ + 1/4⁵ – 1/5⁵ + 1/7⁵ – 1/8⁵ + 1/10⁵ – 1/11⁵ + …

= 32/33 * 3125/3126 * 16807/16806 * 161051/161052 * …

= 4π⁵/(729√3).

An interesting connection can be made here with the Riemann zeta function (excluding all terms where 3 is a factor).

So when s is a positive odd integer,

(1 – 1/2^s + 1/4^s – 1/5^s + 1/7^s – 1/8^s + …)²/(1 + 1/2^2s + 1/4^2s + 1/5^2s + 1/7^2s + 1/8^2s + …)

= 1 – 2/2^s + 2/4^s – 2/5^s + 2/7^s – 2/8^s + 4/10^s – …,

with a product over the primes expression,

= 1/{1 + 2/(2^s +1)} * 1/{1 + 2/(5^s +1)} * 1/{1 – 2/(7^s +1)} * 1/{1 – 2/(11^s +1)} * …

And this result is a rational number.

So when s = 1,

(1 – 1/2 + 1/4 – 1/5 + 1/7 – 1/8 + …)²/(1 + 1/2² + 1/4² + 1/5² + 1/7² + 1/8² + …)

= 1 – 2/2 + 2/4 – 2/5 + 2/7 – 2/8 + 4/10 – …

with a product over primes expression,

1/{1 + 2/(2 + 1)} * 1/{1 + 2/(5 + 1)} * 1/{1 – 2/(7 + 1)} * 1/{1 – 2/(11 + 1)} * …

=  3/5 * 6/8 * 8/6 * 10/12 * 14/16 * 18/20 * …

= 1/4.

And when s = 3,

(1 – 1/2³ + 1/4³ – 1/5³ + 1/7³ – 1/8³ + …)²/(1 + 1/2⁶ + 1/4⁶ + 1/5⁶+ 1/7⁶ + 1/8⁶ + …)

= 1 – 2/2³ + 2/4³ – 2/5³ + 2/7³ – 2/8³ + 4/10³ – …,

with a product over primes expression,

1/{1 + 2/(2³  + 1)} * 1/{1 + 2/(5³  + 1)} * 1/{1 – 2/(7³  + 1)} * 1/{1 – 2/(11³  + 1)} * …

= 9/11 * 126/128 * 344/342 * 1332/1330 *

= 10/13.

Finally briefly when s = 5 

(1 – 1/2⁵ + 1/4⁵ – 1/5⁵ + 1/7⁵ – 1/8⁵ + …)²/(1 + 1/2¹⁰ + 1/4¹⁰+ 1/5¹⁰+ 1/7¹⁰ + 1/8¹⁰ + …)

= 1 – 2/2⁵ + 2/4⁵ – 2/5⁵ + 2/7⁵ – 2/8⁵ + 4/10⁵ – …  = 6930/7381.