Description of Tables

In this webpage, you will find three tables of moments. The first three tables correspond to the normalized central moments of \(S_{\pi}(T^{\alpha},T,\gamma)\) for \(L(s,\pi)\) corresponding to \(\zeta(s), L(s,\Delta)\) and \(L(s,E).\) These moments are normalized by the standard deviation. We expect these moments to follow the moments of a random variable \(X\) that follows normal distribution with mean zero and standard deviation described below: \begin{align*} \text{E}(X^k) =\begin{cases} 0 & \quad \text{ if } k \text{ is odd}\\ (k-1)!! &\quad \text{ o.w.} \end{cases} \end{align*} For example, 1,4, and 15 are second, fourth and sixth moments, respectively.

\(\zeta(s)\): Central Moments of \( \mathcal{S}(T^{\alpha},T,\gamma)\)

\(\alpha\)	Mean	SD	Third	Fourth	Fifth	Sixth
0.01	8.12664	0.71270	-1.67873	7.35662	-30.92888	160.08244
0.1	0.84594	0.48606	-0.14800	2.63000	-1.20722	10.33348
0.2	0.27746	0.71235	0.01296	2.68812	0.10514	10.89217
0.25	0.30232	0.80378	0.04913	2.80095	0.41680	12.30439
0.5	0.62593	1.16800	0.02198	2.87361	0.23124	13.19703
0.75	0.93729	1.42760	-0.02490	2.82371	-0.23509	12.56396
1	1.02474	1.48873	0.00424	2.85382	0.04322	12.96062
1.5	0.99992	1.47122	-0.00414	2.84552	-0.03014	12.86350
2	0.99701	1.46975	-0.00297	2.85074	-0.03193	12.88880

\(L(s,\Delta)\): Central Moments of \( \mathcal{S}_{\Delta}(T^{2\alpha},T,\gamma)\)

\(\alpha\)	Mean	SD	Third	Fourth	Fifth	Sixth
0.01	11.98130	1.07698	-1.85257	7.98510	-35.14171	186.15826
0.1	0.21967	0.65852	-0.03054	2.69843	-0.25359	11.03453
0.25	0.31208	1.19386	0.02998	2.95844	0.28935	14.43506
0.5	0.64105	1.69936	0.00284	2.92765	0.04688	13.95892
0.75	0.95958	2.06712	-0.00476	2.91615	-0.03678	13.77586
1	0.99514	2.11018	0.00382	2.95486	0.02674	14.32046
1.5	1.00261	2.12086	-0.00301	2.93006	-0.04713	13.89995
2	1.00055	2.11442	-0.00495	2.91746	-0.04286	13.81671

\(L(s,E)\): Central Moments of \( \mathcal{S}_{E}(T^{2\alpha},T,\gamma)\)

\(\alpha\)	Mean	SD	Third	Fourth	Fifth	Sixth
0.01	14.05733	1.14519	-1.68426	7.48952	-32.11737	171.47675
0.1	0.23650	0.72693	-0.06557	2.19741	-0.35269	6.71268
0.25	0.24893	1.15014	0.01560	2.96012	0.14544	14.35478
0.5	0.54386	1.68869	0.02265	2.98378	0.22220	14.79037
0.75	0.81496	2.08139	-0.00486	2.93901	-0.03015	14.11834
1	1.01970	2.33100	-0.00259	2.92996	-0.05075	13.93498
1.5	0.99791	2.29736	0.00446	2.95346	0.03002	14.27238
2	0.99767	2.30323	0.00187	2.94319	0.02775	14.21050