L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (0.959 − 0.281i)5-s + (0.415 + 0.909i)6-s + (0.654 + 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)10-s + (−0.841 − 0.540i)11-s + (0.841 + 0.540i)12-s + (−0.654 + 0.755i)13-s + (0.959 + 0.281i)14-s + (0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.415 − 0.909i)17-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (0.959 − 0.281i)5-s + (0.415 + 0.909i)6-s + (0.654 + 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)10-s + (−0.841 − 0.540i)11-s + (0.841 + 0.540i)12-s + (−0.654 + 0.755i)13-s + (0.959 + 0.281i)14-s + (0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.415 − 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.083085753 - 0.1672266969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.083085753 - 0.1672266969i\) |
\(L(1)\) |
\(\approx\) |
\(1.683741036 - 0.1252447533i\) |
\(L(1)\) |
\(\approx\) |
\(1.683741036 - 0.1252447533i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 11 | \( 1 + (-0.841 - 0.540i)T \) |
| 13 | \( 1 + (-0.654 + 0.755i)T \) |
| 17 | \( 1 + (-0.415 - 0.909i)T \) |
| 19 | \( 1 + (-0.415 + 0.909i)T \) |
| 29 | \( 1 + (0.415 + 0.909i)T \) |
| 31 | \( 1 + (-0.142 - 0.989i)T \) |
| 37 | \( 1 + (0.959 + 0.281i)T \) |
| 41 | \( 1 + (-0.959 + 0.281i)T \) |
| 43 | \( 1 + (0.142 - 0.989i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.654 + 0.755i)T \) |
| 59 | \( 1 + (-0.654 + 0.755i)T \) |
| 61 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (-0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.841 - 0.540i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (0.654 - 0.755i)T \) |
| 83 | \( 1 + (0.959 + 0.281i)T \) |
| 89 | \( 1 + (0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.959 - 0.281i)T \) |
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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
−39.33482589629605334274113911229, −37.17654685153411594795120987538, −36.165657406880078154000483734925, −34.52186655028900444597055748881, −33.7368874145980544708760013327, −32.415047556969178546750986775205, −30.69616045890646108967806673004, −30.04958408653655997840449446567, −28.821038780962283511499050632948, −26.37364047768445097234497531551, −25.24581029958346973264782227137, −24.10433004235077031212481342476, −23.05211363346169218849256179827, −21.58826294349130332800064351903, −20.10092556287774272054688421257, −17.81718023429885344931949177912, −17.218062968277537192783677955175, −14.9267297498883991927605622614, −13.64357095106010376302118665678, −12.7144367997136803495128723353, −10.80333450597320314622754371375, −7.96669685476324552466378490646, −6.682122929086607135531999587529, −5.12010715496190400374515675553, −2.35411168788392419350667347488,
2.473712372400737954111502514698, 4.7463531721381654522177335807, 5.80015224206707440634641050469, 9.14729345802198967749456421469, 10.52705860820053999574560484821, 11.9600848508228963369632335014, 13.79395777511727040962942646010, 14.99691157482120151679073361859, 16.509337093457073672242413117537, 18.4276888619449538673689442386, 20.50746819469770569999969825314, 21.38467987556147591535047687090, 22.146580197257299225445900613472, 23.91896783424091893298494423174, 25.28896530532217233251280210055, 27.157031368684222388671121074695, 28.56662802267029037447180766571, 29.324395814452884084665761854139, 31.32375436378571313955777553146, 32.03221983563972448910809793722, 33.53372883901863086772922396536, 34.05165362945192283689215830256, 36.7946839313765054068397415341, 37.63438665308647743595581587069, 38.74397818811945791663464534861