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
−25.06963411138544231983486947400, −24.05912688577765347661741388617, −23.09314422543480153221558669449, −22.34342617762518754435363180163, −21.5245826644709017870980186705, −20.18836616169984764592659903997, −19.79844018022450078581514960240, −18.56159443244106809309010347006, −17.793611948987790785831873965710, −16.72589488468043137115504291004, −15.73417812590505537203573451369, −15.10715470091901487078611947990, −13.74518612751725299895316529715, −12.93370879325680026021805046510, −12.1369242093520169225796862471, −10.85260757535825954864655109560, −9.87595920907272074090070173086, −9.13926579995790251964582111595, −7.67449663664633135437784140404, −6.93953408156690245361187866078, −5.64780050451695938864429881578, −4.64297445497966664634012445880, −3.20794221004802115128672263340, −2.25534286602273722048272878357, −0.12421076659372906193632066592,
1.989674628569118491646590549049, 3.20779457345998553494177489687, 4.32877037691338478273057914305, 5.73096603622603057252817710577, 6.55337091280841884711051415504, 7.77915171531275978380878355755, 8.79554169796547434622782728792, 9.96080610090000950124595333862, 10.63316995807809364041672046462, 12.02826983279790294513547895975, 12.780860071119646392943897333188, 13.74410677524177807051657375227, 14.76385578750301259005646216581, 15.86887205482668471875300964542, 16.54625457539077954368199636474, 17.51857895716470358309409201793, 18.8198689233891534494839703804, 19.241516812328445910230181233769, 20.33495990684589206882107159252, 21.39987402752342491901694616540, 22.13043273045746032928303621376, 23.08669823084017557400585137539, 23.980704341807558578151358218678, 24.80533817755050501575063417893, 26.01946188665161814684136078985