L(s) = 1 | + (0.299 + 0.953i)2-s + (−0.406 + 0.913i)3-s + (−0.820 + 0.572i)4-s + (−0.993 − 0.113i)6-s + (−0.791 − 0.610i)8-s + (−0.669 − 0.743i)9-s + (−0.189 − 0.981i)12-s + (−0.389 − 0.921i)13-s + (0.345 − 0.938i)16-s + (−0.263 − 0.964i)17-s + (0.508 − 0.861i)18-s + (−0.625 − 0.780i)19-s + (−0.618 + 0.786i)23-s + (0.879 − 0.475i)24-s + (0.761 − 0.647i)26-s + (0.951 − 0.309i)27-s + ⋯ |
L(s) = 1 | + (0.299 + 0.953i)2-s + (−0.406 + 0.913i)3-s + (−0.820 + 0.572i)4-s + (−0.993 − 0.113i)6-s + (−0.791 − 0.610i)8-s + (−0.669 − 0.743i)9-s + (−0.189 − 0.981i)12-s + (−0.389 − 0.921i)13-s + (0.345 − 0.938i)16-s + (−0.263 − 0.964i)17-s + (0.508 − 0.861i)18-s + (−0.625 − 0.780i)19-s + (−0.618 + 0.786i)23-s + (0.879 − 0.475i)24-s + (0.761 − 0.647i)26-s + (0.951 − 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.07137120767 + 0.6704233615i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.07137120767 + 0.6704233615i\) |
\(L(1)\) |
\(\approx\) |
\(0.5924003122 + 0.5225626779i\) |
\(L(1)\) |
\(\approx\) |
\(0.5924003122 + 0.5225626779i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.299 + 0.953i)T \) |
| 3 | \( 1 + (-0.406 + 0.913i)T \) |
| 13 | \( 1 + (-0.389 - 0.921i)T \) |
| 17 | \( 1 + (-0.263 - 0.964i)T \) |
| 19 | \( 1 + (-0.625 - 0.780i)T \) |
| 23 | \( 1 + (-0.618 + 0.786i)T \) |
| 29 | \( 1 + (0.998 + 0.0570i)T \) |
| 31 | \( 1 + (-0.483 + 0.875i)T \) |
| 37 | \( 1 + (-0.956 - 0.290i)T \) |
| 41 | \( 1 + (-0.198 - 0.980i)T \) |
| 43 | \( 1 + (0.281 + 0.959i)T \) |
| 47 | \( 1 + (-0.508 - 0.861i)T \) |
| 53 | \( 1 + (0.938 - 0.345i)T \) |
| 59 | \( 1 + (-0.948 + 0.318i)T \) |
| 61 | \( 1 + (0.217 + 0.976i)T \) |
| 67 | \( 1 + (-0.0950 - 0.995i)T \) |
| 71 | \( 1 + (0.774 + 0.633i)T \) |
| 73 | \( 1 + (0.244 - 0.969i)T \) |
| 79 | \( 1 + (0.432 + 0.901i)T \) |
| 83 | \( 1 + (0.441 + 0.897i)T \) |
| 89 | \( 1 + (-0.888 + 0.458i)T \) |
| 97 | \( 1 + (0.999 + 0.0285i)T \) |
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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
−18.38497172002120663041277842624, −17.28416233740188811411248460301, −17.077830006607752627886043618941, −16.08995780317161713635857123927, −15.010978214351722308318445945896, −14.31288386383608170325256581467, −13.8784202816108584506215884198, −13.00161559589659019520018283485, −12.520896212397877705854581838517, −11.94927074109858948987378427039, −11.34782077007642334651913142413, −10.54180489146094364176805909545, −10.06866832819940743540208145732, −8.98871665680585093893498385686, −8.39340428402432042425091066285, −7.66707123302254328899289417059, −6.44529323399747002001172894978, −6.24664690164560961455368855971, −5.23503016406762647310458627918, −4.4570362783221997757880179144, −3.77905339020792219115117384950, −2.672775237632797980989391339197, −1.97581166769379516611144858628, −1.46476908221919174643820336079, −0.24482033706623887938421656062,
0.74281431327563441765405285416, 2.48201456625235654128930211690, 3.308382038258130891345182779808, 3.967489575259809224742715238093, 5.01737382543093897933255037215, 5.09893711020495045129971661887, 6.0478905651766534769857786030, 6.79664762157162789717540946894, 7.473099504802431833205400301335, 8.44234555196322847733508592520, 8.9689698622800623612592951305, 9.73243938659307765610391111188, 10.371618519787690576462212622209, 11.18434833654929675726146412614, 12.079873942154655172270424015761, 12.54607726542418490468202259511, 13.59941053525153462230557681774, 14.04299115032915447810678068036, 14.98166858011921515801660636560, 15.36950031080301357925955817634, 15.98981997983691254889277035405, 16.512789668540174703704079433500, 17.37913759738551193288900681357, 17.76522424944646229482073966412, 18.284855207915664719673662620372