L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.921 + 0.388i)3-s + (0.623 + 0.781i)4-s + (−0.988 − 0.149i)5-s + (−0.661 − 0.749i)6-s + (−0.318 + 0.947i)7-s + (−0.222 − 0.974i)8-s + (0.698 + 0.715i)9-s + (0.826 + 0.563i)10-s + (−0.411 + 0.911i)11-s + (0.270 + 0.962i)12-s + (0.980 − 0.198i)13-s + (0.698 − 0.715i)14-s + (−0.853 − 0.521i)15-s + (−0.222 + 0.974i)16-s + (−0.853 + 0.521i)17-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.921 + 0.388i)3-s + (0.623 + 0.781i)4-s + (−0.988 − 0.149i)5-s + (−0.661 − 0.749i)6-s + (−0.318 + 0.947i)7-s + (−0.222 − 0.974i)8-s + (0.698 + 0.715i)9-s + (0.826 + 0.563i)10-s + (−0.411 + 0.911i)11-s + (0.270 + 0.962i)12-s + (0.980 − 0.198i)13-s + (0.698 − 0.715i)14-s + (−0.853 − 0.521i)15-s + (−0.222 + 0.974i)16-s + (−0.853 + 0.521i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6188731543 + 0.3978670923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6188731543 + 0.3978670923i\) |
\(L(1)\) |
\(\approx\) |
\(0.7591101350 + 0.1673713573i\) |
\(L(1)\) |
\(\approx\) |
\(0.7591101350 + 0.1673713573i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 127 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 3 | \( 1 + (0.921 + 0.388i)T \) |
| 5 | \( 1 + (-0.988 - 0.149i)T \) |
| 7 | \( 1 + (-0.318 + 0.947i)T \) |
| 11 | \( 1 + (-0.411 + 0.911i)T \) |
| 13 | \( 1 + (0.980 - 0.198i)T \) |
| 17 | \( 1 + (-0.853 + 0.521i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.411 - 0.911i)T \) |
| 29 | \( 1 + (0.456 + 0.889i)T \) |
| 31 | \( 1 + (0.878 - 0.478i)T \) |
| 37 | \( 1 + (0.173 + 0.984i)T \) |
| 41 | \( 1 + (0.878 + 0.478i)T \) |
| 43 | \( 1 + (-0.998 - 0.0498i)T \) |
| 47 | \( 1 + (0.365 - 0.930i)T \) |
| 53 | \( 1 + (0.270 - 0.962i)T \) |
| 59 | \( 1 + (-0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.955 - 0.294i)T \) |
| 67 | \( 1 + (-0.124 - 0.992i)T \) |
| 71 | \( 1 + (0.995 + 0.0995i)T \) |
| 73 | \( 1 + (0.0747 - 0.997i)T \) |
| 79 | \( 1 + (-0.583 + 0.811i)T \) |
| 83 | \( 1 + (-0.969 + 0.246i)T \) |
| 89 | \( 1 + (0.826 - 0.563i)T \) |
| 97 | \( 1 + (0.542 + 0.840i)T \) |
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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
−28.52118378685378373518043207773, −27.242919445924208860435824611, −26.49255033207031542260101398102, −26.021215350490332343252908471633, −24.710193465459574578049930807958, −23.79442128065633210468229942468, −23.19451477505623321040428262128, −21.14025472514991697959378389206, −20.02462181531663958471949022535, −19.46476358194799061958948707825, −18.63783102019949665326828736167, −17.5143576208218859159017991742, −15.941424727609940111315889274107, −15.64242354016392683334705747915, −14.144736834068700703547692426200, −13.309570607180880517245164948085, −11.5051175462261017982463085113, −10.5753893200969891841937610307, −9.12124651541949505896886708360, −8.22737775576919455836504406151, −7.33658227364265372764217012344, −6.39045385848413931757626467200, −4.14030321567365941469178894856, −2.78382473272142851125839193756, −0.84090234052337298140982073798,
1.99452227633004037862958525132, 3.22089795652773049220475457666, 4.38866776802933857370216813783, 6.65891232448323210547653209606, 8.21240373656736933699211781378, 8.49710533898879503333143169481, 9.81933125264673133169909696185, 10.85874664060425625417832680121, 12.219266119367674239571488765639, 13.02705491992012994130905879668, 15.01094965093631946166839372798, 15.60495598631365188085434399393, 16.4784688205812524215987720870, 18.20887694359637434518826474687, 18.88658137606269710415793879035, 19.909981218041273216947490639689, 20.56069464170162612464254389380, 21.581802260295925232423748484676, 22.79947573103377934480114050268, 24.382256460171036238977111028836, 25.3935042499399365373230525475, 26.07829073632388380045350135795, 27.06441004464613341342134522140, 28.02862385696136120324399134514, 28.50544838123733517189876754064