L(s) = 1 | + (0.978 − 0.207i)2-s + (−0.913 − 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.309 − 0.951i)5-s + (−0.978 − 0.207i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s − 12-s + (−0.104 + 0.994i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.809 + 0.587i)18-s + (−0.104 − 0.994i)19-s + (−0.669 − 0.743i)20-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (−0.913 − 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.309 − 0.951i)5-s + (−0.978 − 0.207i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s − 12-s + (−0.104 + 0.994i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.809 + 0.587i)18-s + (−0.104 − 0.994i)19-s + (−0.669 − 0.743i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2406791569 - 1.468405577i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2406791569 - 1.468405577i\) |
\(L(1)\) |
\(\approx\) |
\(1.040649581 - 0.7222728314i\) |
\(L(1)\) |
\(\approx\) |
\(1.040649581 - 0.7222728314i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (-0.978 - 0.207i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.104 - 0.994i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.104 + 0.994i)T \) |
| 41 | \( 1 + (0.913 + 0.406i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (-0.978 - 0.207i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.978 - 0.207i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.669 - 0.743i)T \) |
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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
−22.007954430468253732912558074853, −21.673058982604400609606507242573, −20.69686768154372283455470890420, −19.82994273458816658352595216482, −18.85506891873967188072349788197, −17.88447571629154083375588599145, −17.24662737353424529070787873550, −16.160665018658817249992879510576, −15.75862949460494538207291384026, −14.88173774128964432346864860615, −14.29668832400870348430118495100, −13.234455612437411005932669736640, −12.35336680031972732501309968676, −11.6641119776997828583965094016, −10.87157660731558131319317652075, −10.4538050280401703443394627184, −9.20638159120747304955419897083, −7.70849448809124238141125259179, −7.13698576413160941120022916497, −6.065473067665486881638913908130, −5.76874521413799731125208909116, −4.39985582646171849916828167632, −3.92332743505534214684560457284, −2.89172777090443031294897719233, −1.68784082476585369764642378395,
0.48992264040575966888805700391, 1.631344668660735162344271522526, 2.628271535663100708284841524695, 4.16212218711759966364521919190, 4.64788788009763211462019183181, 5.45422746792018564235398930262, 6.35528345321023289948325195911, 7.093901419057830936162323382798, 8.08733167322150923528251734590, 9.23208328839910830315354014127, 10.428813277883399170995226348431, 11.13372562047035407465645053683, 11.96289950108940612648163581595, 12.42199406512187134094164591839, 13.324510883024122309390394839935, 13.73721370638108987090503473589, 15.14871135844281837748748932034, 15.7934472808085750632779780271, 16.46668605787960320040689202322, 17.229376407953554421841195714213, 18.109829623047266225819816472568, 19.21355772339323533496789220029, 19.8396267893488171789876166502, 20.60670159118061884183444167870, 21.484399772521451267848637667951