L(s) = 1 | + (−0.649 + 0.760i)2-s + (−0.155 − 0.987i)4-s + (−0.694 − 0.719i)5-s + (0.833 + 0.552i)7-s + (0.852 + 0.523i)8-s + (0.998 − 0.0611i)10-s + (0.993 + 0.115i)11-s + (0.352 + 0.935i)13-s + (−0.961 + 0.275i)14-s + (−0.951 + 0.307i)16-s + (−0.989 + 0.142i)17-s + (−0.359 + 0.933i)19-s + (−0.601 + 0.798i)20-s + (−0.733 + 0.680i)22-s + (−0.0339 + 0.999i)25-s + (−0.940 − 0.339i)26-s + ⋯ |
L(s) = 1 | + (−0.649 + 0.760i)2-s + (−0.155 − 0.987i)4-s + (−0.694 − 0.719i)5-s + (0.833 + 0.552i)7-s + (0.852 + 0.523i)8-s + (0.998 − 0.0611i)10-s + (0.993 + 0.115i)11-s + (0.352 + 0.935i)13-s + (−0.961 + 0.275i)14-s + (−0.951 + 0.307i)16-s + (−0.989 + 0.142i)17-s + (−0.359 + 0.933i)19-s + (−0.601 + 0.798i)20-s + (−0.733 + 0.680i)22-s + (−0.0339 + 0.999i)25-s + (−0.940 − 0.339i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0412 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0412 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8657630866 + 0.9022851875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8657630866 + 0.9022851875i\) |
\(L(1)\) |
\(\approx\) |
\(0.7602910971 + 0.2968014158i\) |
\(L(1)\) |
\(\approx\) |
\(0.7602910971 + 0.2968014158i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.649 + 0.760i)T \) |
| 5 | \( 1 + (-0.694 - 0.719i)T \) |
| 7 | \( 1 + (0.833 + 0.552i)T \) |
| 11 | \( 1 + (0.993 + 0.115i)T \) |
| 13 | \( 1 + (0.352 + 0.935i)T \) |
| 17 | \( 1 + (-0.989 + 0.142i)T \) |
| 19 | \( 1 + (-0.359 + 0.933i)T \) |
| 31 | \( 1 + (0.980 - 0.195i)T \) |
| 37 | \( 1 + (0.639 - 0.768i)T \) |
| 41 | \( 1 + (0.0950 + 0.995i)T \) |
| 43 | \( 1 + (0.980 + 0.195i)T \) |
| 47 | \( 1 + (-0.563 - 0.826i)T \) |
| 53 | \( 1 + (0.377 + 0.925i)T \) |
| 59 | \( 1 + (0.723 + 0.690i)T \) |
| 61 | \( 1 + (-0.268 - 0.963i)T \) |
| 67 | \( 1 + (-0.115 - 0.993i)T \) |
| 71 | \( 1 + (-0.262 - 0.965i)T \) |
| 73 | \( 1 + (0.728 - 0.685i)T \) |
| 79 | \( 1 + (0.977 - 0.209i)T \) |
| 83 | \( 1 + (-0.314 + 0.949i)T \) |
| 89 | \( 1 + (0.994 - 0.101i)T \) |
| 97 | \( 1 + (-0.384 - 0.923i)T \) |
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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
−17.574307987447309593186397328750, −17.33506925719058704952323265832, −16.28205801107912233073503073219, −15.64826411131530183787753007117, −14.964207502630577015072498123760, −14.22637825220035057007143189507, −13.51963496377446846575485035570, −12.89290399923793767569782205865, −11.918472903933721378415172562725, −11.47735518054114648293282634643, −10.937066398547622741824727915572, −10.48593938132058664662547550863, −9.67477802983252423232261103409, −8.724995521890279800459558709401, −8.34257986159999171381138168432, −7.58718452907957069866947175162, −6.9428047091674356521654715025, −6.367652841534345973410776514433, −5.01265606615308560861006646588, −4.20965434982349415200049919390, −3.83894699313817033724644609008, −2.8738734624973620390105402197, −2.30019340760442942247346103774, −1.1971780495368257663285408374, −0.544355041809653796156589435337,
0.8688948062841493828137070814, 1.61389222683394073561189082829, 2.22875018417746209954000160021, 3.7873446153430402670194872341, 4.444049241418312689621532616, 4.84004324469313935504931191285, 5.936479612886896061214580626658, 6.366248485526555530443119858698, 7.24585153943939321242275863852, 8.01770982473987339435682144748, 8.45404517591978145537255038454, 9.143598798232264833548520344633, 9.46909423461574217344281284202, 10.66133234004448349079560953469, 11.29209981630344185214562998543, 11.81007282965152530529958835585, 12.495846972314986101796818046431, 13.50438671667417731116567484464, 14.14751616077744768828198481005, 14.850817566739378081050034932642, 15.277708024285064750342437252005, 16.0060051751549971269617359798, 16.6977409515541963652208535868, 17.00898685851405958265589836242, 17.90637510602514152211075978987