L(s) = 1 | − 7-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.809 + 0.587i)23-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.809 + 0.587i)41-s − 43-s + (0.309 − 0.951i)47-s + 49-s + (−0.309 + 0.951i)53-s + (−0.809 − 0.587i)59-s + ⋯ |
L(s) = 1 | − 7-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.809 + 0.587i)23-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.809 + 0.587i)41-s − 43-s + (0.309 − 0.951i)47-s + 49-s + (−0.309 + 0.951i)53-s + (−0.809 − 0.587i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03926706968 - 0.2058451879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03926706968 - 0.2058451879i\) |
\(L(1)\) |
\(\approx\) |
\(0.6519498525 - 0.06586564388i\) |
\(L(1)\) |
\(\approx\) |
\(0.6519498525 - 0.06586564388i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.809 + 0.587i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.309 - 0.951i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−26.01946188665161814684136078985, −24.80533817755050501575063417893, −23.980704341807558578151358218678, −23.08669823084017557400585137539, −22.13043273045746032928303621376, −21.39987402752342491901694616540, −20.33495990684589206882107159252, −19.241516812328445910230181233769, −18.8198689233891534494839703804, −17.51857895716470358309409201793, −16.54625457539077954368199636474, −15.86887205482668471875300964542, −14.76385578750301259005646216581, −13.74410677524177807051657375227, −12.780860071119646392943897333188, −12.02826983279790294513547895975, −10.63316995807809364041672046462, −9.96080610090000950124595333862, −8.79554169796547434622782728792, −7.77915171531275978380878355755, −6.55337091280841884711051415504, −5.73096603622603057252817710577, −4.32877037691338478273057914305, −3.20779457345998553494177489687, −1.989674628569118491646590549049,
0.12421076659372906193632066592, 2.25534286602273722048272878357, 3.20794221004802115128672263340, 4.64297445497966664634012445880, 5.64780050451695938864429881578, 6.93953408156690245361187866078, 7.67449663664633135437784140404, 9.13926579995790251964582111595, 9.87595920907272074090070173086, 10.85260757535825954864655109560, 12.1369242093520169225796862471, 12.93370879325680026021805046510, 13.74518612751725299895316529715, 15.10715470091901487078611947990, 15.73417812590505537203573451369, 16.72589488468043137115504291004, 17.793611948987790785831873965710, 18.56159443244106809309010347006, 19.79844018022450078581514960240, 20.18836616169984764592659903997, 21.5245826644709017870980186705, 22.34342617762518754435363180163, 23.09314422543480153221558669449, 24.05912688577765347661741388617, 25.06963411138544231983486947400