L(s) = 1 | + (−0.267 − 0.963i)2-s + (−0.222 + 0.974i)3-s + (−0.857 + 0.514i)4-s + (−0.958 − 0.283i)5-s + (0.998 − 0.0460i)6-s + (0.407 − 0.913i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (−0.996 + 0.0804i)11-s + (−0.311 − 0.950i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.490 − 0.871i)15-s + (0.469 − 0.882i)16-s + (0.00575 − 0.999i)17-s + ⋯ |
L(s) = 1 | + (−0.267 − 0.963i)2-s + (−0.222 + 0.974i)3-s + (−0.857 + 0.514i)4-s + (−0.958 − 0.283i)5-s + (0.998 − 0.0460i)6-s + (0.407 − 0.913i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (−0.996 + 0.0804i)11-s + (−0.311 − 0.950i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.490 − 0.871i)15-s + (0.469 − 0.882i)16-s + (0.00575 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4532192829 + 0.2121925704i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4532192829 + 0.2121925704i\) |
\(L(1)\) |
\(\approx\) |
\(0.5952111275 - 0.07757516741i\) |
\(L(1)\) |
\(\approx\) |
\(0.5952111275 - 0.07757516741i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 \) |
good | 2 | \( 1 + (-0.267 - 0.963i)T \) |
| 3 | \( 1 + (-0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.958 - 0.283i)T \) |
| 7 | \( 1 + (0.407 - 0.913i)T \) |
| 11 | \( 1 + (-0.996 + 0.0804i)T \) |
| 13 | \( 1 + (0.365 + 0.930i)T \) |
| 17 | \( 1 + (0.00575 - 0.999i)T \) |
| 19 | \( 1 + (-0.806 - 0.591i)T \) |
| 23 | \( 1 + (0.641 + 0.767i)T \) |
| 29 | \( 1 + (0.386 - 0.922i)T \) |
| 31 | \( 1 + (0.256 + 0.966i)T \) |
| 37 | \( 1 + (-0.244 + 0.969i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.998 - 0.0575i)T \) |
| 47 | \( 1 + (0.948 + 0.316i)T \) |
| 53 | \( 1 + (-0.999 + 0.0115i)T \) |
| 59 | \( 1 + (-0.0402 + 0.999i)T \) |
| 61 | \( 1 + (0.658 + 0.752i)T \) |
| 67 | \( 1 + (-0.667 + 0.744i)T \) |
| 71 | \( 1 + (0.993 - 0.114i)T \) |
| 73 | \( 1 + (0.490 + 0.871i)T \) |
| 79 | \( 1 + (0.449 + 0.893i)T \) |
| 83 | \( 1 + (-0.632 - 0.774i)T \) |
| 89 | \( 1 + (0.322 + 0.946i)T \) |
| 97 | \( 1 + (-0.684 + 0.729i)T \) |
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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
−23.54206538290617590290620593876, −22.80749129481351933229315349367, −21.979423231051004155677886367696, −20.62253894040464793426379129714, −19.45745254089344385743008388339, −18.737488584329421838134510005636, −18.3598433177041810291402448679, −17.468325815048143778064887612548, −16.5410999933635020240961060706, −15.472319180183756623787089480294, −15.02602269236207466491921484908, −14.09010011280343079062066600814, −12.73755535231949520337650243594, −12.55938651884144226335106158776, −11.06273208607220946152234176928, −10.46329830190385300206939852639, −8.633143377485253397496406221658, −8.26904173211328946563649928099, −7.58027525460249834524337830892, −6.50529142481100998200576342896, −5.70804428926284261226451895690, −4.83286270377631294754243171811, −3.33822045016201434798559524539, −1.97278921489216761783015105119, −0.37359174454378545319136923401,
1.01724059426156170305475546676, 2.73850202893578235904985400875, 3.70166747062585016715104936457, 4.59388997196374834357683283239, 5.00038040013097434029996419713, 6.959594315921505442856832269000, 8.06519768014988963719952581279, 8.80270248677850479500402920187, 9.81257332614047100823984522847, 10.67361143863203312730225901617, 11.34784284000746207768648911540, 11.89432136297893277437373939825, 13.2124222642141330430805991399, 13.95332466475169227581871703586, 15.13848824255888349608999294500, 16.02104094565908573959968967026, 16.809496155371339152892108995014, 17.54880857001202729143050080691, 18.62861219652221517613047923571, 19.54591777037063406164683463633, 20.28957662652032285297655663368, 20.96974792368764770148728847799, 21.45282911108557068622001023971, 22.66399674389889361793712111477, 23.43163220964205547279533826196