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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09052162189 - 0.05071434356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09052162189 - 0.05071434356i\) |
\(L(1)\) |
\(\approx\) |
\(0.6358607692 + 0.4540727649i\) |
\(L(1)\) |
\(\approx\) |
\(0.6358607692 + 0.4540727649i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−22.29567844719641662211266210612, −21.5727536352501128809267739136, −20.51424586303718816209851553655, −19.83469273137685117195000837418, −19.29458366047894567104336355242, −18.84750992517736074308346238409, −17.35709380270108398531089189202, −16.68959145888541851908532825057, −15.62654714376701946546624071719, −14.99781420789664592077644760015, −14.00852746626440013669125186488, −13.19922862527259113799839892837, −12.4574257524021215114476101662, −11.734914788887697385199166074923, −11.02126540715285697662356861734, −10.02940900703073334042171727649, −9.233774844860303242218803538008, −8.37191941069251113132825943166, −7.01289360030174393899095582402, −6.3434737541186246580811437354, −5.03951141310332274065071589359, −4.19611840556414579042203171512, −3.546297458841793257073469592241, −2.567331155298338162053495434326, −1.15464235845553914440556495973,
0.04219650045041716881365930878, 2.42010295156675898278965184765, 3.26625696261632147775813424394, 4.28160331283068429516282613116, 4.91517964479406220819464187692, 6.35584048035190313433494832223, 6.801146541030231282316701541010, 7.563726992846697181797162946156, 8.775428662083665464394362483002, 9.23494404510957028772914705794, 10.62787750588607384277275760818, 11.6610939384546656407285875556, 12.56211806178466029159694184863, 12.8203854096153113889231326159, 14.24754464960501016573324149298, 14.83615612408903991615487655075, 15.53148757767008685434238633478, 16.13153252566435385106351052676, 17.13213670590108325503764173709, 17.736160361428059128527607557855, 18.96994825880279226368957485577, 19.51091625630147207841047342284, 20.35339725605381193502861646622, 21.677055141974897057075821763843, 22.229998177450852714688037678454