L(s) = 1 | + (−0.944 + 0.327i)2-s + (−0.222 − 0.974i)3-s + (0.785 − 0.618i)4-s + (0.998 − 0.0460i)5-s + (0.529 + 0.848i)6-s + (0.983 + 0.183i)7-s + (−0.539 + 0.842i)8-s + (−0.900 + 0.433i)9-s + (−0.928 + 0.370i)10-s + (−0.200 + 0.979i)11-s + (−0.778 − 0.627i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.267 − 0.963i)15-s + (0.233 − 0.972i)16-s + (−0.991 + 0.126i)17-s + ⋯ |
L(s) = 1 | + (−0.944 + 0.327i)2-s + (−0.222 − 0.974i)3-s + (0.785 − 0.618i)4-s + (0.998 − 0.0460i)5-s + (0.529 + 0.848i)6-s + (0.983 + 0.183i)7-s + (−0.539 + 0.842i)8-s + (−0.900 + 0.433i)9-s + (−0.928 + 0.370i)10-s + (−0.200 + 0.979i)11-s + (−0.778 − 0.627i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.267 − 0.963i)15-s + (0.233 − 0.972i)16-s + (−0.991 + 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.000036957 - 0.4227524220i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.000036957 - 0.4227524220i\) |
\(L(1)\) |
\(\approx\) |
\(0.8513463476 - 0.1769668307i\) |
\(L(1)\) |
\(\approx\) |
\(0.8513463476 - 0.1769668307i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 \) |
good | 2 | \( 1 + (-0.944 + 0.327i)T \) |
| 3 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.998 - 0.0460i)T \) |
| 7 | \( 1 + (0.983 + 0.183i)T \) |
| 11 | \( 1 + (-0.200 + 0.979i)T \) |
| 13 | \( 1 + (0.365 - 0.930i)T \) |
| 17 | \( 1 + (-0.991 + 0.126i)T \) |
| 19 | \( 1 + (0.211 - 0.977i)T \) |
| 23 | \( 1 + (0.924 - 0.381i)T \) |
| 29 | \( 1 + (0.770 + 0.636i)T \) |
| 31 | \( 1 + (-0.832 + 0.553i)T \) |
| 37 | \( 1 + (-0.667 - 0.744i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.300 - 0.953i)T \) |
| 47 | \( 1 + (0.692 - 0.721i)T \) |
| 53 | \( 1 + (0.968 + 0.250i)T \) |
| 59 | \( 1 + (-0.632 + 0.774i)T \) |
| 61 | \( 1 + (0.993 + 0.114i)T \) |
| 67 | \( 1 + (0.932 - 0.359i)T \) |
| 71 | \( 1 + (-0.819 + 0.572i)T \) |
| 73 | \( 1 + (-0.267 + 0.963i)T \) |
| 79 | \( 1 + (0.675 + 0.736i)T \) |
| 83 | \( 1 + (0.799 - 0.600i)T \) |
| 89 | \( 1 + (-0.596 - 0.802i)T \) |
| 97 | \( 1 + (0.641 - 0.767i)T \) |
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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
−23.572254626846005836109505773034, −22.178940593553531889064018746733, −21.54794520820614563163295012294, −20.930200451022780462126659487231, −20.49200495730472823337059166841, −19.20130925317892302773886669875, −18.28857137796642021843972159943, −17.55543568227767018759690653696, −16.82954387497770626928806817058, −16.23611144282382970407182030093, −15.1671587728344172775815193792, −14.18941816712787701755112893025, −13.33489667629271069641730852728, −11.80687261752436563186256569496, −11.12688212981806833928009543330, −10.566889936463689767112879301804, −9.566483391549379754456682086006, −8.886773913260118937673517367575, −8.110696890350481995907395412334, −6.65946730279498387794667924065, −5.8019985814179500599565649746, −4.66256834064567322230289936424, −3.48441785054651881392746854945, −2.321843188779523937206624233801, −1.19763254178379303115081020953,
0.9428994208244991578131889051, 1.961902441945234231402639102266, 2.58540409255459112289542805584, 5.07080039563315390256156252300, 5.55839321270188347300399562210, 6.85922404665832054495100646106, 7.24947307485227111926210645998, 8.59270535557912262150356324631, 8.94197529850673033415721353204, 10.4840409829828814888568094299, 10.87828603176089337101575191133, 12.076978323237441031675997329657, 12.985229087555105336354777944665, 13.949565333671425251893451020810, 14.84413952622453086488795309035, 15.687740931250035191223950723997, 17.0820100491891156537744899549, 17.60999567033415098555811682318, 17.97951768863756259968841086948, 18.6827194632045699920390867121, 20.004649771469091866272039548804, 20.36999546233726140191548197895, 21.47743399373243945124824272000, 22.60692178614813414686080038773, 23.61250855259853599645113496249