L(s) = 1 | + (−0.809 + 0.587i)3-s − 7-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.309 − 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s − 7-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.309 − 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6860868050 - 0.2716409826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6860868050 - 0.2716409826i\) |
\(L(1)\) |
\(\approx\) |
\(0.6814058510 + 0.07659837243i\) |
\(L(1)\) |
\(\approx\) |
\(0.6814058510 + 0.07659837243i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (-0.309 - 0.951i)T \) |
| 67 | \( 1 + (-0.809 - 0.587i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.907098482824209371178760932184, −25.66682772646407448652951735472, −24.81651437625586583531248089583, −23.916592605829102554102459627271, −22.94674680834010112566954539873, −22.2345373860722292347318253919, −21.37435888424012671988829229976, −19.649091314200291735584226907164, −19.299338345887763312989432602177, −18.073796583941104309591082577058, −17.18532970241603721276233468171, −16.31458501508164167395627697048, −15.37381823254475236346257861900, −13.83154359550680770659144476708, −12.95718510657323198512186802891, −12.19538113293942731914972964528, −10.97624159066657532212541060328, −10.15285252246408814557110688527, −8.706765877371283957971846331272, −7.50665550608588924350592105713, −6.286528051360041551960199058615, −5.720839284720482537365765982683, −4.081049392230957735098674625951, −2.61959204790307374855118323208, −0.91674788837959236660351520464,
0.36662623518164172743536726122, 2.40344475569585094997539836301, 4.06664121275498940259478793478, 4.82951134926387359103690088829, 6.44640655356585506865450204559, 6.86898901385480687064989915748, 8.88964936592152505579734742480, 9.7308501465721051223107856890, 10.6233834898413395527458419020, 11.87880225448825717449256085510, 12.556384495775575992116485667075, 13.88763347462555129128394338606, 15.222632334536307364482586924173, 15.93288503722467483487062645462, 16.92602978924085534978880507598, 17.67698930853866646790042660409, 18.902800412505651026853253307035, 19.91781802464345830480172394049, 20.98641315514544683332839454155, 22.04171482624800808507860329267, 22.65574759056682106239831203277, 23.48980959307803587926947436327, 24.59733734282868088013148455846, 25.846305122687593705659018652848, 26.53841919788634403076412866962