L(s) = 1 | + (0.994 − 0.104i)2-s + (0.309 + 0.951i)3-s + (0.978 − 0.207i)4-s + (0.994 + 0.104i)5-s + (0.406 + 0.913i)6-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.5 + 0.866i)12-s + (0.207 + 0.978i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.743 + 0.669i)18-s + (−0.951 + 0.309i)19-s + (0.994 − 0.104i)20-s + ⋯ |
L(s) = 1 | + (0.994 − 0.104i)2-s + (0.309 + 0.951i)3-s + (0.978 − 0.207i)4-s + (0.994 + 0.104i)5-s + (0.406 + 0.913i)6-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.5 + 0.866i)12-s + (0.207 + 0.978i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.743 + 0.669i)18-s + (−0.951 + 0.309i)19-s + (0.994 − 0.104i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.358361862 + 1.811257026i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.358361862 + 1.811257026i\) |
\(L(1)\) |
\(\approx\) |
\(2.336178369 + 0.7014298496i\) |
\(L(1)\) |
\(\approx\) |
\(2.336178369 + 0.7014298496i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.994 - 0.104i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.994 + 0.104i)T \) |
| 17 | \( 1 + (-0.104 + 0.994i)T \) |
| 19 | \( 1 + (-0.951 + 0.309i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.978 - 0.207i)T \) |
| 31 | \( 1 + (-0.994 + 0.104i)T \) |
| 37 | \( 1 + (-0.743 + 0.669i)T \) |
| 41 | \( 1 + (0.743 + 0.669i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.743 + 0.669i)T \) |
| 53 | \( 1 + (-0.104 - 0.994i)T \) |
| 59 | \( 1 + (-0.207 - 0.978i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.406 - 0.913i)T \) |
| 73 | \( 1 + (0.743 - 0.669i)T \) |
| 79 | \( 1 + (-0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.587 - 0.809i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.994 + 0.104i)T \) |
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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
−21.549954517519819127955044346335, −20.83196433114747258273838962075, −20.12463154566141449717802399971, −19.33321563002938067168268466335, −18.406388199973506862479893109858, −17.496915409097670673131871514389, −16.960943524475626475173935032263, −15.866600893256983756972467604785, −14.94279565180487452237534595782, −14.078298490215646666152789846425, −13.691078614335810920788224986644, −12.891580596150331830359765968540, −12.31254850049991650635281429949, −11.34989417108893135617483853330, −10.490462525456841996444977954708, −9.26215752653271629237689618625, −8.47549344675013434464767259719, −7.18169589910774999597177957659, −6.8633622861089625801588962895, −5.75842668052515254938560405406, −5.24304980005269724152458495031, −3.9461892201024505204427994520, −2.77246910344024856860493677447, −2.20333591036992942170791875000, −1.17950845912255391028016043122,
1.649837262558309595519005032145, 2.52256267405440556019666650815, 3.375239571341518877002890213743, 4.38113943375001362901489730229, 5.05493526727613974672222805379, 6.035813631380719055317159668852, 6.59545585677129050371257210699, 8.0265246917584282830162809876, 8.92780032799379973056868150451, 9.995425770209739935419543487546, 10.55278109344743066444341789662, 11.18882579716809400958636008016, 12.47314655586612955523423115330, 13.11333123067165897710081599528, 14.01692883086744300422956869190, 14.65457165399941589433429763857, 15.146209487285227796773057742418, 16.21420559897240015199262696988, 16.82312792045449718686521198767, 17.60166955852113947333573962892, 18.94332666597753361710826590903, 19.69092896390806362505705308270, 20.60063748362542693242581093276, 21.169071352273881249036472399376, 21.68284974310930173548697938665