L(s) = 1 | + (0.342 + 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.642 + 0.766i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (0.642 − 0.766i)33-s + (−0.939 − 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.642 + 0.766i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (0.642 − 0.766i)33-s + (−0.939 − 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6159335717 - 0.3579684895i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6159335717 - 0.3579684895i\) |
\(L(1)\) |
\(\approx\) |
\(0.8427613172 + 0.1541694611i\) |
\(L(1)\) |
\(\approx\) |
\(0.8427613172 + 0.1541694611i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (0.342 + 0.939i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.642 + 0.766i)T \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.984 + 0.173i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.984 + 0.173i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.642 - 0.766i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−20.28728533803584287040930029071, −20.17666790996186023009075019297, −19.3245731864464337088444906053, −18.34370439744249477267767487445, −18.14183907696456433331408296584, −17.14369680604329073107667832658, −16.23597066091161101056420743009, −15.502293216860341359639780521593, −14.62465455572008577793808654637, −13.91994773496835332030582963865, −13.058793096049806022073530336025, −12.43177034113880911824182082468, −12.034801247740305890875435446061, −10.8198463705875283023789359459, −9.65393531004841043198336822556, −9.473093518079088641001049751423, −8.14862369547385369288068325333, −7.40996284403393871530049555768, −7.00978241363732358099838807514, −5.75941451320407959919660003756, −5.3071255211582273455987996260, −3.77317535351268289286620494813, −2.95605336437897718346730092952, −2.26520659264773977942193360001, −1.0742886463794833383867702841,
0.27163384985469823596560218118, 2.043710782893452011618456538707, 3.038285925125493855977124903766, 3.66184219084908118105397046284, 4.523029234812560228205749978436, 5.58221272524564562143929332672, 6.187501876339401960356164098421, 7.42196133937980928584626181235, 8.14919239691615879398092346006, 9.2085500070352068603877274150, 9.6398507102812916617089113989, 10.453486050361681336553138166040, 11.16476992223818300116675744505, 12.128520123522566902802191580487, 13.01590222209872800647452529664, 13.934244340491106908856191082947, 14.36347967629797035400222692821, 15.45182097132117385356756242051, 16.00744961546000825521595453602, 16.6471881884016428752224068102, 17.2018885623587223747355957670, 18.55550390121327215226089215897, 19.10928521233931551896904213332, 19.860353580983360528495571334668, 20.49696102897369278691284084755