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[parent] values of the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ at negative integers (Example)

As an example of the parent entry, we exhibit a few values of the Dedekind zeta function $ \zeta(s,K)$ associated to the number field $ K=\mathbb{Q}(\sqrt{5})$. The following are the first few values of $ \zeta(1-k,K)$ for $ k\geq 2$ even ( $ \zeta(1-k,K)=0$ for odd $ k$):

$ k$ $ \zeta(1-k,K)$
$ 2$ $ 1/30$
$ 4$ $ 1/60$
$ 6$ $ 67/630$
$ 8$ $ 361/120$
$ 10$ $ 412751/1650$
$ 12$ $ 795286411/16380$
$ 14$ $ 568591843/30$
$ 16$ $ 54701427071177/4080$
$ 18$ $ 571363169189645713/35910$
$ 20$ $ 98510726938027364651/3300$
$ 22$ $ 58282448789678207092153/690$
$ 24$ $ 11364600197977872303826339891/32760$
$ 26$ $ 60097244486962154421889002337/30$
$ 28$ $ 27553534229181632149212403498558667/1740$
$ 30$ $ 179897691732312705009829008469165679351567/1074150$
$ 32$ $ 18959952687425212997198188226124948760804457/8160$
$ 34$ $ 1246857198273009466623646468891606914655865453/30$
$ 36$ $ 32613724432434114882079789497713006473885167100345949413/34545420$
$ 38$ $ 806046617423971060143933083079087480165364733020665319/30$
$ 40$ $ 256143677142597053855778958609943339670281438740737609308546491/270600$
$ 42$ $ 7744103059592598379932370402622990883812122603998003590194837592521/189630$
$ 44$ $ 2948486831915768046445018911155037347932306554653422623861453575795893/1380$
$ 46$ $ 189448047879825594335580602295205397993735968958688272066548049250586050109/1410$
$ 48$ $ 11220849734737403891785284646197712257525539376899788451604867938805644674521295667/1113840$
$ 50$ $ 7374830701193124787713243044624034212252297046043499899645907770420650321652018100351/8250$



"values of the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ at negative integers" is owned by alozano.
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Cross-references: odd, even, number field, Dedekind zeta function

This is version 2 of values of the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ at negative integers, born on 2006-06-20, modified 2006-06-20.
Object id is 8065, canonical name is ValuesOfTheDedekindZetaFunctionOfMathbbQsqrt5AtNegativeIntegers.
Accessed 719 times total.

Classification:
AMS MSC: 11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)
 11R42 (Number theory :: Algebraic number theory: global fields :: Zeta functions and $L$-functions of number fields)
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