L(s) = 1 | + (0.990 − 0.139i)2-s + (0.961 − 0.275i)4-s + (0.766 − 0.642i)5-s + (0.559 − 0.829i)7-s + (0.913 − 0.406i)8-s + (0.669 − 0.743i)10-s + (0.997 + 0.0697i)11-s + (−0.990 − 0.139i)13-s + (0.438 − 0.898i)14-s + (0.848 − 0.529i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.559 − 0.829i)20-s + (0.997 − 0.0697i)22-s + (−0.559 − 0.829i)23-s + ⋯ |
L(s) = 1 | + (0.990 − 0.139i)2-s + (0.961 − 0.275i)4-s + (0.766 − 0.642i)5-s + (0.559 − 0.829i)7-s + (0.913 − 0.406i)8-s + (0.669 − 0.743i)10-s + (0.997 + 0.0697i)11-s + (−0.990 − 0.139i)13-s + (0.438 − 0.898i)14-s + (0.848 − 0.529i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.559 − 0.829i)20-s + (0.997 − 0.0697i)22-s + (−0.559 − 0.829i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.925126874 - 4.335020902i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.925126874 - 4.335020902i\) |
\(L(1)\) |
\(\approx\) |
\(2.158843762 - 1.055450832i\) |
\(L(1)\) |
\(\approx\) |
\(2.158843762 - 1.055450832i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.990 - 0.139i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.559 - 0.829i)T \) |
| 11 | \( 1 + (0.997 + 0.0697i)T \) |
| 13 | \( 1 + (-0.990 - 0.139i)T \) |
| 17 | \( 1 + (0.104 - 0.994i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.559 - 0.829i)T \) |
| 29 | \( 1 + (-0.990 + 0.139i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.882 + 0.469i)T \) |
| 43 | \( 1 + (0.374 - 0.927i)T \) |
| 47 | \( 1 + (0.0348 + 0.999i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.374 - 0.927i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (-0.104 + 0.994i)T \) |
| 73 | \( 1 + (0.104 + 0.994i)T \) |
| 79 | \( 1 + (0.719 - 0.694i)T \) |
| 83 | \( 1 + (-0.990 + 0.139i)T \) |
| 89 | \( 1 + (0.104 + 0.994i)T \) |
| 97 | \( 1 + (-0.997 - 0.0697i)T \) |
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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
−22.10446832135658857042799735293, −21.5191209105180413913709077559, −21.099826710870723948248527925553, −19.769426933229909551245988552157, −19.22558713624780923529544116628, −18.08602681830482924239221815292, −17.19502105511326425194232333336, −16.70182766236308454463195123227, −15.27251227847531823391137404866, −14.80453311488766319184010511978, −14.32314766325423374859822462095, −13.3471518300322150942735037042, −12.45169892167008741013362209948, −11.72773594216056474643667748461, −10.93630127912580031206858176230, −9.98282006978731562813869246320, −8.96435260718954362733318266015, −7.84059978195464899111717119950, −6.883007104700562690922096071237, −6.02116871829599270037192724129, −5.484596159212913841169632909998, −4.32585119879562376559024721686, −3.40729078969367686530595574159, −2.13782969704523850007946691513, −1.803929675000508075280664969682,
0.70749129725814981767054540436, 1.73080045440343861343081650814, 2.60233691740393027373151518019, 4.02132776372869246144232760164, 4.63470684233640733289470683021, 5.419419787894955117673228902531, 6.504974188990274059236470842749, 7.196002204082412981038290388805, 8.32340562737019519480719999138, 9.542834542942789933714794397006, 10.23053199525175077995013539270, 11.22933891135329643574348450129, 12.05259370022228372994838504010, 12.82132230281869034213893961045, 13.6408003602061672081614059814, 14.34203156845253117522740818649, 14.849897821237676898599862034139, 16.11648845922746071455266150547, 17.01419208830476757281726803956, 17.21955101673092463449337706594, 18.5909912464759309604277421348, 19.729229566769762560238077959889, 20.3727756026087444168021141863, 20.77919837104529523470145329574, 21.92698144707355604670590987058