L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·8-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s − i·22-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (0.866 − 0.5i)32-s + 34-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·8-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s − i·22-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (0.866 − 0.5i)32-s + 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9627814922 + 0.3086037039i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9627814922 + 0.3086037039i\) |
\(L(1)\) |
\(\approx\) |
\(0.7598669203 + 0.008769799075i\) |
\(L(1)\) |
\(\approx\) |
\(0.7598669203 + 0.008769799075i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + iT \) |
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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
−29.06490993082341868909198829502, −28.31056643161893585527916343241, −27.03033361068190413371345094413, −26.505160921884719109851635668942, −25.270038429522025431868621806358, −24.368210293070398263207030129490, −23.52178434263885797583643547908, −22.10012493336450873553113608460, −20.85995943378496989635583665717, −19.57281905143085665840039199458, −18.891198241868669739163879207228, −17.667197671753166942708064863006, −16.710741351085207103886921399838, −15.780488957520345088551310329, −14.58505839338957233853180210190, −13.511181624682297645329643005422, −11.65250498436462286564549982025, −10.8133713934863976178472766919, −9.30398791532513874994872140242, −8.60229030709830511192760034388, −7.074146408136281702781573802136, −6.17494625695281691109265234437, −4.55649631400347342231553276808, −2.43090943168424051338935901478, −0.656763106144636492254270513824,
1.28545574320467412604837383845, 2.814021635486487853837418196362, 4.344009238310901585269287945343, 6.346832000547836373518674817279, 7.62495496162157788404088425380, 8.75296833606427697630851110535, 9.94537300750917393211273626622, 10.899527875158376994517040206831, 12.19698044363250037505479412661, 13.06968959711227095685536012436, 14.84033213196302041630689640948, 15.91975728076383453758620820007, 17.28402208602199572639927448465, 17.838613327444404705347714105241, 19.20454056702653794192165610150, 20.01005422237504138210518029260, 20.99395131501536992557818102316, 22.123095672788914329762838243593, 23.226138856447726290735585731529, 24.91635464617152454946938763908, 25.42140627891992476643154942547, 26.73563917612246869275601172413, 27.53188900891677466114426735352, 28.45153276131990912215210992429, 29.44500922570862087760624308814