L(s) = 1 | + (0.207 − 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (−0.743 − 0.669i)13-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (0.587 + 0.809i)18-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯ |
L(s) = 1 | + (0.207 − 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (−0.743 − 0.669i)13-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (0.587 + 0.809i)18-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8039032500 - 1.145438281i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8039032500 - 1.145438281i\) |
\(L(1)\) |
\(\approx\) |
\(1.069605821 - 0.2419033885i\) |
\(L(1)\) |
\(\approx\) |
\(1.069605821 - 0.2419033885i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.207 - 0.978i)T \) |
| 3 | \( 1 + (0.406 + 0.913i)T \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 13 | \( 1 + (-0.743 - 0.669i)T \) |
| 17 | \( 1 + (0.743 - 0.669i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (-0.913 - 0.406i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (-0.994 - 0.104i)T \) |
| 53 | \( 1 + (0.743 + 0.669i)T \) |
| 59 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + (0.978 - 0.207i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.669 - 0.743i)T \) |
| 73 | \( 1 + (0.994 - 0.104i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (0.951 + 0.309i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.207 + 0.978i)T \) |
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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
−21.61660125040034579987783266566, −20.8991642700911619112624149651, −19.80991454066970247957962633743, −19.24107007011954238432836279597, −18.11559752650316527813171904717, −17.8327764219154485195753248879, −16.73581947335544503824815203508, −16.42175473792150718381863145533, −14.84466052727023508605888186867, −14.574003267642458351853590295008, −13.90668379227086463669568192391, −13.03695517683539982072110829982, −12.37603102700936669117891652015, −11.47370487309868758968860041399, −10.19188157658729722967356546187, −9.24975708686224323414467323624, −8.31475084522237929498646695288, −7.68556315865774139255293674678, −7.05814577238633137876603242023, −6.24930467902829677218883965117, −5.26957155178628680825482759600, −4.228598206515491800952640949342, −3.40928600003603800028571737026, −2.03272494262965805167029529257, −0.947307339921074105234035357317,
0.2851990932523439427775987188, 1.86428656762481656657919832923, 2.6264753456354674286525136169, 3.462769727836205402984464466486, 4.42983586553080667377446339309, 5.2781669126935527268408254317, 5.77306981627169968404410142406, 7.737433799743288810226864587627, 8.31749025748920235082947020590, 9.481327468248740672134056386116, 9.71120992456919561555362937825, 10.74601342735107434301110421829, 11.55708394288347824981341761902, 12.1310404349868127430219259145, 13.21231526049419454902772554372, 14.04123015705678753926368803415, 14.87433917280849307581493720339, 15.21323141993701706596364405953, 16.39491758039977111732687643655, 17.33713828719823294144091840381, 18.17695076299861347051461328282, 18.94794102291873184222694225754, 19.76209777933774031179104537620, 20.554068038085733978917469313544, 20.95455608244677452337495235348