L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s − i·23-s + 27-s − i·29-s − i·31-s + (0.5 − 0.866i)33-s + (0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s − i·23-s + 27-s − i·29-s − i·31-s + (0.5 − 0.866i)33-s + (0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3855803695 - 0.1099828306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3855803695 - 0.1099828306i\) |
\(L(1)\) |
\(\approx\) |
\(0.5939327497 + 0.1921892885i\) |
\(L(1)\) |
\(\approx\) |
\(0.5939327497 + 0.1921892885i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 - iT \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.866 - 0.5i)T \) |
| 61 | \( 1 + (-0.866 - 0.5i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.866 + 0.5i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 - iT \) |
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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
−20.61096113388037522712142407787, −19.852648967250699591659534735, −19.14659234578167846826262822591, −18.52797852855354588559820696188, −17.73631390947914323312797219565, −16.90296280060955598211381789296, −16.53559742667364345485295574527, −15.54499018844179484014998294292, −14.52693515763480253842406232853, −13.6924604232273845988494156469, −13.08296249985344483354986659679, −12.50774470133811461216214177418, −11.57846634796909388646806887019, −10.77830470107614988062237587401, −10.15067241659868414323038508561, −9.151759223061945563686734361938, −7.95332884221418745483972409489, −7.514889512255274055435601452226, −6.5721042468104735358387953755, −6.05476846570183248142221621819, −4.8181271096593746099332653958, −4.2283160128352187628308900321, −2.696668471702350485525252633780, −2.17123744507742114042659012952, −0.747732679430399409879929557439,
0.22003203272803682752313057993, 2.10873638807366402671713618318, 2.931774335543905711024748564044, 3.834495628938095273045881052928, 5.01520146953331800143635437013, 5.36199726646740404677214438322, 6.32837449451264779788916109580, 7.1985447901931358182729100030, 8.44054563375828708170577649003, 9.04468630926865737595958514825, 9.887752268364864593372980780374, 10.62799791176041756658814892377, 11.2103153362051251522801750103, 12.42231854687999273701169741845, 12.62793893705029614924646088134, 13.79154277571384599030094781009, 14.92866906279857131404240597251, 15.43073678606550815601254956954, 15.86312383602141372512362555179, 16.82074472246602840784764421771, 17.62538146846000089705903151133, 18.09994397119574901384513445549, 19.24302515472753756992374474204, 19.79713606832715536778370982459, 20.7739943174966125464393882649