L(s) = 1 | + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (−0.809 − 0.587i)10-s + (−0.848 − 0.529i)11-s + (−0.997 − 0.0697i)13-s + (0.882 + 0.469i)14-s + (−0.719 − 0.694i)16-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (−0.0348 + 0.999i)20-s + (0.0348 + 0.999i)22-s + (−0.848 + 0.529i)23-s + ⋯ |
L(s) = 1 | + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (−0.809 − 0.587i)10-s + (−0.848 − 0.529i)11-s + (−0.997 − 0.0697i)13-s + (0.882 + 0.469i)14-s + (−0.719 − 0.694i)16-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (−0.0348 + 0.999i)20-s + (0.0348 + 0.999i)22-s + (−0.848 + 0.529i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6389635131 + 0.1800915108i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6389635131 + 0.1800915108i\) |
\(L(1)\) |
\(\approx\) |
\(0.6462398644 - 0.2355681585i\) |
\(L(1)\) |
\(\approx\) |
\(0.6462398644 - 0.2355681585i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.559 - 0.829i)T \) |
| 5 | \( 1 + (0.939 - 0.342i)T \) |
| 7 | \( 1 + (-0.882 + 0.469i)T \) |
| 11 | \( 1 + (-0.848 - 0.529i)T \) |
| 13 | \( 1 + (-0.997 - 0.0697i)T \) |
| 17 | \( 1 + (0.978 - 0.207i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (-0.848 + 0.529i)T \) |
| 29 | \( 1 + (-0.559 - 0.829i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.241 + 0.970i)T \) |
| 43 | \( 1 + (0.559 + 0.829i)T \) |
| 47 | \( 1 + (0.241 - 0.970i)T \) |
| 53 | \( 1 + (0.978 - 0.207i)T \) |
| 59 | \( 1 + (0.997 + 0.0697i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.766 + 0.642i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (-0.374 - 0.927i)T \) |
| 83 | \( 1 + (-0.438 + 0.898i)T \) |
| 89 | \( 1 + (-0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.882 + 0.469i)T \) |
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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
−22.20152972507065065967630304168, −20.99757003753570296851003742313, −20.19659469690620151215159314626, −19.136128484756845051771416720852, −18.66521894226344136818983353632, −17.73219176174497641351567525278, −17.01196278435166207572397805252, −16.475387586181026507261749417305, −15.49965490102197857516085456678, −14.582725550487839256467077505123, −14.05872183853270245590687389794, −13.048322752067625710652921240906, −12.36575158019328541483407655484, −10.599081386119200935084070923437, −10.18812754643450919291699962021, −9.657170979585905170828790446772, −8.5999250781574306708114756978, −7.474430331132637234197396092676, −6.93303073118852696347489729043, −5.94091010876102706186797822078, −5.30835922514566891470607238735, −4.10546072967438535552461848735, −2.67278285670485850999437115731, −1.68602871157952148072624334364, −0.23193375232157142186158845011,
0.75541919181080836982526984336, 2.226798320844214913334232723052, 2.65522794306982598883246274420, 3.82889853926029792525744089842, 5.13073855717589213617181857815, 5.87763284597959190514212747960, 7.11558862110271338040575832234, 8.13422980126446889747023070773, 9.00277371216340483072996242954, 9.878268337343412746588574679991, 10.13589690683768049296149621173, 11.355792753105183429488808952726, 12.31029034688194032055984108326, 12.96402361833471116365815671051, 13.52946436151330888761194234532, 14.6125008921026938428536971370, 15.91630346089752460629877699829, 16.555980251428920699649856315, 17.36141938019034559463490215181, 18.08500528153098556650203040822, 18.943788736418152506502229462157, 19.49826459303135702522671072445, 20.42630082951681598362479617219, 21.37446124966571221770947331563, 21.63944427209485785630929940099