L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.258 + 0.965i)3-s + (−0.669 − 0.743i)4-s + (−0.207 − 0.978i)5-s + (−0.987 − 0.156i)6-s + (0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (0.978 + 0.207i)10-s + (−0.838 + 0.544i)11-s + (0.544 − 0.838i)12-s + (−0.156 + 0.987i)13-s + (0.891 − 0.453i)15-s + (−0.104 + 0.994i)16-s + (−0.544 − 0.838i)17-s + (−0.104 − 0.994i)18-s + (−0.777 + 0.629i)19-s + ⋯ |
L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.258 + 0.965i)3-s + (−0.669 − 0.743i)4-s + (−0.207 − 0.978i)5-s + (−0.987 − 0.156i)6-s + (0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (0.978 + 0.207i)10-s + (−0.838 + 0.544i)11-s + (0.544 − 0.838i)12-s + (−0.156 + 0.987i)13-s + (0.891 − 0.453i)15-s + (−0.104 + 0.994i)16-s + (−0.544 − 0.838i)17-s + (−0.104 − 0.994i)18-s + (−0.777 + 0.629i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1097137808 + 0.2419361870i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1097137808 + 0.2419361870i\) |
\(L(1)\) |
\(\approx\) |
\(0.4611128433 + 0.3715229467i\) |
\(L(1)\) |
\(\approx\) |
\(0.4611128433 + 0.3715229467i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.406 + 0.913i)T \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 + (-0.207 - 0.978i)T \) |
| 11 | \( 1 + (-0.838 + 0.544i)T \) |
| 13 | \( 1 + (-0.156 + 0.987i)T \) |
| 17 | \( 1 + (-0.544 - 0.838i)T \) |
| 19 | \( 1 + (-0.777 + 0.629i)T \) |
| 23 | \( 1 + (-0.913 - 0.406i)T \) |
| 29 | \( 1 + (0.453 + 0.891i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.978 + 0.207i)T \) |
| 43 | \( 1 + (-0.587 - 0.809i)T \) |
| 47 | \( 1 + (0.358 + 0.933i)T \) |
| 53 | \( 1 + (0.998 + 0.0523i)T \) |
| 59 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 + (0.994 + 0.104i)T \) |
| 67 | \( 1 + (0.0523 - 0.998i)T \) |
| 71 | \( 1 + (-0.891 - 0.453i)T \) |
| 73 | \( 1 + (0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.965 - 0.258i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.629 + 0.777i)T \) |
| 97 | \( 1 + (-0.891 + 0.453i)T \) |
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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
−25.1953021244193283057373162405, −23.85873702753199977635173498772, −23.135553395263000825712626816472, −22.144565826638977270348033651725, −21.32081087018997095418472423108, −20.046504782660543163646554494678, −19.47677968936799149594054263121, −18.66980679334659882069193795605, −17.91620972177448783157676502913, −17.30375282243148644819871711968, −15.659165497571732171382759833681, −14.55299388824674942142810508194, −13.46026465669837757643732645020, −12.87803403975400043480198705456, −11.751775999900016312843685463488, −10.87731889658584103735816307456, −10.10794594543023878916201334293, −8.56130328811808428742573447853, −7.95671366191927154650991946775, −6.92973600332364691187763239542, −5.64068105374714593475412839435, −3.8052447993067846299282054195, −2.81264349464204494211376253067, −2.00622056147039777765786945678, −0.18305217346598808751040019197,
2.016564071549955269422715145646, 4.04629139657935046454689613164, 4.76509641071875095645874612315, 5.619599829044113406929511902399, 7.09537368963249176228636448833, 8.26412403429606400176473550596, 8.94065907542182351716809507057, 9.798073661552479045056690894862, 10.72901436973999336897529217838, 12.158429506772263110783722007499, 13.43356747240216562605229474105, 14.34015440795954495748858499958, 15.32194810480399990840072944718, 16.14261626056454128433711119574, 16.59114284474679429236154743655, 17.62109122788951347046293215570, 18.7407452895997500341021035699, 19.87016520538478350186943896066, 20.56423476993343238702722967735, 21.57545015727630314761946719854, 22.65553744798833801904572493197, 23.600430395436024398053965495506, 24.32579421551761314529345077670, 25.44858072822850320659259354775, 25.98965574265559300283224686062