L(s) = 1 | + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + i·5-s + (−0.5 + 0.866i)6-s + 7-s − 8-s + 9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s − i·15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + i·5-s + (−0.5 + 0.866i)6-s + 7-s − 8-s + 9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s − i·15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2064648100 + 0.3788431460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2064648100 + 0.3788431460i\) |
\(L(1)\) |
\(\approx\) |
\(0.8087624042 - 0.1533308047i\) |
\(L(1)\) |
\(\approx\) |
\(0.8087624042 - 0.1533308047i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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
−17.91738649568932274794403237167, −17.44944248929455450358481274857, −16.79203964713580650340991405752, −16.28196761033547799982802422809, −15.58912964527038894005720878708, −15.09171995427400336586912583756, −14.06762383351283272223933339376, −13.42780831172890742752875018140, −12.780030326157834298078806197718, −12.25083015476031664362368665490, −11.35136931137669900282816860528, −11.07632481339807997006522018600, −9.697356720738901980080479232495, −9.1922391023244790541754625282, −8.2265891892991057437992133730, −7.640310665019041003011673074743, −7.018134705635114411120210262467, −6.11211401663289103048180151277, −5.18857499278913818371463632866, −4.82760649460136085069064898960, −4.712411072181174603408662396371, −3.33003067695517496535880240276, −2.29065886248929441245040020438, −1.04757967559122730143037446670, −0.13227888951156300984528307535,
1.24327126722078923152895815862, 2.08797209763841375960950606311, 2.65509999449634970227501591905, 3.85027950141714895007468373844, 4.4479902874451051078368250869, 5.24818261208963309561888782280, 5.72156433552670777768835253775, 6.62712059325081785214765433730, 7.43582597427165902297820110185, 8.08075453327610379875726807749, 9.46102756389301382788387268544, 10.020132421621880268523740136564, 10.74614208335211410299591552037, 11.02649831484433397769417296128, 11.79320235514312063635894252634, 12.379601868412904834446061399492, 13.05972641789823181910943392691, 13.86913614436529658028030024514, 14.66214248800735804231276855945, 15.165130669339822028548053654433, 15.66055220776645275995422944236, 17.02511061409190234731269877120, 17.50822609573510412879071712662, 18.05217436912259611691823262411, 18.81038199925044117327608759361