L(s) = 1 | + (0.743 − 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 − 0.994i)4-s + (0.743 + 0.669i)5-s + (0.207 − 0.978i)6-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + 10-s + (−0.5 − 0.866i)12-s + (0.994 + 0.104i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.406 − 0.913i)18-s + (−0.587 − 0.809i)19-s + (0.743 − 0.669i)20-s + ⋯ |
L(s) = 1 | + (0.743 − 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 − 0.994i)4-s + (0.743 + 0.669i)5-s + (0.207 − 0.978i)6-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + 10-s + (−0.5 − 0.866i)12-s + (0.994 + 0.104i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.406 − 0.913i)18-s + (−0.587 − 0.809i)19-s + (0.743 − 0.669i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.872488849 - 2.820287537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.872488849 - 2.820287537i\) |
\(L(1)\) |
\(\approx\) |
\(1.778489658 - 1.306700805i\) |
\(L(1)\) |
\(\approx\) |
\(1.778489658 - 1.306700805i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.743 - 0.669i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.743 + 0.669i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.587 - 0.809i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.104 + 0.994i)T \) |
| 31 | \( 1 + (-0.743 + 0.669i)T \) |
| 37 | \( 1 + (0.406 + 0.913i)T \) |
| 41 | \( 1 + (-0.406 + 0.913i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.406 - 0.913i)T \) |
| 53 | \( 1 + (0.669 + 0.743i)T \) |
| 59 | \( 1 + (0.994 + 0.104i)T \) |
| 61 | \( 1 + (-0.309 - 0.951i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.207 + 0.978i)T \) |
| 73 | \( 1 + (-0.406 - 0.913i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.951 + 0.309i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.743 + 0.669i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.772152820739030553749490556689, −20.99475685546493055762668913927, −20.8938381470363629513020955506, −19.77019120040485504784144107507, −18.87659546423681013483853232974, −17.63580074914552456269330036488, −16.92713661069514477604281490989, −16.33079739423422932911090859808, −15.51162466882471301915605351983, −14.64102501705491279842009507917, −14.2027760168455042565923910826, −13.13626977335802421330848542860, −12.889539853069452100427884203589, −11.706380580936786501362949684276, −10.520413406182038125049608012164, −9.610587082316448964829123658634, −8.839250714773961770387362079689, −8.08515795423200311648815542632, −7.31671130360301302779601525175, −5.91403257034717179874064487373, −5.50089774754676946219966657067, −4.33847570143869037002088566139, −3.78863224697101228895588277644, −2.62400074956362727739198595917, −1.70806652227336867637858685272,
1.044391875600966845648892407, 2.06990073390089260185032502196, 2.81415440365290138321228550143, 3.45033088918210671658005044975, 4.70946424038714748987090455583, 5.69481063920535414659233076854, 6.72843748046374162576797259017, 7.14873402320862617219253984655, 8.64225219857779511677021115401, 9.387619298686950454797711149913, 10.2255430618550845116677480673, 11.00329557151441510120739169024, 12.01241135676977009832246976006, 12.842191488008833980229402876546, 13.46796381762353105432633574967, 14.193573742194181610069405642437, 14.73677088871735975249943180527, 15.42802760629782556308897587005, 16.71294304375235857826606836777, 17.93533014949605917713766112450, 18.5252092916874284668919311480, 19.051152017877021587192492458690, 20.05670983728141668016548707874, 20.55170568053887355890261012989, 21.48133728141076557125257775011