Spectral parameter of Maass forms of level 1
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Numbers
$R'$ 
$R$
9.533:
9.533695261353557554344235235928770323821256395107251982375790464135348991298347781769255509975435366
12.17:
12.17300832467967784952795117639554812398247167309994790041359894085944536082660887402607610119914083
13.77:
13.77975135189073894424367328151771259715513256879348706925238822161445033353997009415783160955742757
14.35:
14.35850951825981277986694256903716549561438589919676624781520226663201120679288581901319549358192409
16.13:
16.13807317152103058019829428598600394563144288541378695827382712175947030542755279355556642723837034
16.64:
16.64425920189981994352627455936865570143168145997231928907651455001829017618970424409102246827670179
17.73:
17.73856338105737789321732636154654617200548005325129188079624689810214157759980377279197640233860653
18.18:
18.18091783453070386031830826819331393824992456541781787106792508774526862910670490089247820557750868
19.42:
19.42348147082825519163378035720852444158552560076333197577947593786262611347059612969474807916047941
19.48:
19.48471385474101336412852642787287621877406238534520661308580751557226039659320657991642152653178001
20.10:
20.1066946826
21.31:
21.3157959402
21.47:
21.4790575447
22.19:
22.1946739776
22.78:
22.7859084942
23.20:
23.2013961812
23.26:
23.2637115379
24.11:
24.1123527298
24.41:
24.4197154423
25.05:
25.0508548508
25.82:
25.8262437127
26.05:
26.0569177607
26.15:
26.1520854492
26.44:
26.446996418
27.28:
27.2843840117
27.33:
27.3327080831
27.77:
27.7759207018
28.51:
28.5102777031
28.53:
28.5307476929
28.86:
28.8633943539
29.13:
29.1375875578
29.54:
29.5463881241
31.52:
31.5265821968
31.56:
31.5662754118
32.50:
32.5081177599
32.89:
32.8911702135
34.02:
34.0278842001
34.45:
34.456271533
35.50:
35.5023497714
35.84:
35.8416764326
36.67:
36.6775529931
36.85:
36.8563494959
37.82:
37.8250722906
38.30:
38.3032761525
39.16:
39.1680849679
39.40:
39.4075318615
39.77:
39.773622619
40.54:
40.5433512105
40.68:
40.6886664449
41.55:
41.5555776736
41.88:
41.8830032854
42.64:
42.6434884147
42.92:
42.9222277836
43.26:
43.2671820388
44.07:
44.0774047617
44.42:
44.4263481186
45.28:
45.2874384425
45.36:
45.3616136021
45.39:
45.3984695313
46.10:
46.1014563216
46.48:
46.4814024123
46.65:
46.65331836
47.42:
47.4228958985
47.92:
47.9265583306
48.03:
48.0393309051
48.74:
48.7416663476
48.99:
48.9983076541
49.68:
49.6835200753
49.96:
49.9616962905
Definition
A Maass form (of weight 0) of level 1 is a smooth, square-integrable, automorphic eigenform of the Laplace-Beltrami operator $\Delta$ that is invariant under $SL(2,\mathbb{Z})$. The Laplace eigenvalue of a Maass form has the form $\lambda = 1/4 + R^2$, where $R > 0$ is called the spectral parameter.
Parameters
$R'$
—   spectral parameter (truncated)
References
[1]
A. Booker, A. Strömbergsson, A. Venkatesh, "Effective Computation of Maass Cusp Forms". IMRN 2006, Article ID 71281, 34 pages. (data files) http://www2.math.uu.se/~astrombe/papers/bsv27april06.pdf
[2]
Strömberg, Fredrik, "Maass waveforms on (Γ0(N),χ) (computational aspects)". Hyperbolic geometry and applications in quantum chaos and cosmology, 187–228, London Math. Soc. Lecture Note Ser., 397, Cambridge Univ. Press, 2012
Links
Data properties
Entries are of type: real number
Table is complete: no
Sources of data: [3] (higher precision available, from Booker, Strömbergsson, Venkatesh [1]), [4] (computed by Fredrik Stroemberg [2] and Holger Then)
Reliability: For heuristics see [3] and [5].