L(s) = 1 | + (0.858 + 0.512i)2-s + (0.753 + 0.657i)3-s + (0.473 + 0.880i)4-s + (0.936 + 0.351i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.0448 + 0.998i)8-s + (0.134 + 0.990i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (−0.550 − 0.834i)13-s + (−0.393 − 0.919i)14-s + (0.473 + 0.880i)15-s + (−0.550 + 0.834i)16-s + (−0.550 + 0.834i)17-s + (−0.393 + 0.919i)18-s + ⋯ |
L(s) = 1 | + (0.858 + 0.512i)2-s + (0.753 + 0.657i)3-s + (0.473 + 0.880i)4-s + (0.936 + 0.351i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.0448 + 0.998i)8-s + (0.134 + 0.990i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (−0.550 − 0.834i)13-s + (−0.393 − 0.919i)14-s + (0.473 + 0.880i)15-s + (−0.550 + 0.834i)16-s + (−0.550 + 0.834i)17-s + (−0.393 + 0.919i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.741324197 + 2.407450858i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.741324197 + 2.407450858i\) |
\(L(1)\) |
\(\approx\) |
\(1.773652750 + 1.272829622i\) |
\(L(1)\) |
\(\approx\) |
\(1.773652750 + 1.272829622i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (0.858 + 0.512i)T \) |
| 3 | \( 1 + (0.753 + 0.657i)T \) |
| 5 | \( 1 + (0.936 + 0.351i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.550 - 0.834i)T \) |
| 17 | \( 1 + (-0.550 + 0.834i)T \) |
| 19 | \( 1 + (0.983 - 0.178i)T \) |
| 23 | \( 1 + (-0.900 + 0.433i)T \) |
| 29 | \( 1 + (0.753 - 0.657i)T \) |
| 31 | \( 1 + (0.858 + 0.512i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (-0.393 - 0.919i)T \) |
| 47 | \( 1 + (0.983 - 0.178i)T \) |
| 53 | \( 1 + (0.936 - 0.351i)T \) |
| 59 | \( 1 + (-0.0448 - 0.998i)T \) |
| 61 | \( 1 + (0.858 - 0.512i)T \) |
| 67 | \( 1 + (-0.900 - 0.433i)T \) |
| 71 | \( 1 + (0.134 - 0.990i)T \) |
| 73 | \( 1 + (-0.963 + 0.266i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.858 - 0.512i)T \) |
| 89 | \( 1 + (-0.222 - 0.974i)T \) |
| 97 | \( 1 + (0.134 + 0.990i)T \) |
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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
−23.70594211926987960986261703156, −22.47549914324195187229278745529, −21.930686904166857743960666039056, −20.97655363996080357592130411502, −20.25699150906251640696429803815, −19.515514287252804996752760842982, −18.62038968972873218112683622389, −17.96276601071448669742711395170, −16.48108967736818506300753766832, −15.63009168009969161165753779743, −14.54430653339550402670918017107, −13.75684662661517456717589986239, −13.3491758772283482025487644545, −12.216485611742539324958905693861, −11.88327099227938927040020728638, −10.11029368227599402165471506477, −9.53267812426908269577417182251, −8.70027659012891077749159096166, −7.05592813245212075706138616932, −6.39969414552703329532088038346, −5.4270481913595224356224279209, −4.25241611069508338637573242825, −2.85643172486628735148619537943, −2.38065208526136743889721862369, −1.20898536226087282162637479817,
2.12228860338864071788744299962, 3.055564753933208504215019976814, 3.82914588195386195870109926469, 5.00333071844843553061661708398, 5.92762358714511640171034440310, 6.9521555248455384387937650986, 7.867578722326185966974396712375, 9.00672459592179779576612897169, 10.09153464305001606888406436622, 10.603822827901171611799965968288, 12.144899595924556913960196042375, 13.26257609823684815569063154125, 13.71276949648558459578826632630, 14.454770567014616512146101120813, 15.46686473093265463877828256327, 15.995658297829896971017959006124, 17.13922583621244562863578520362, 17.70694200253833400387628459640, 19.30448000294268863596948204143, 20.08597291200862650436181605321, 20.82298256351587136228810097036, 21.821269508606812286968878504421, 22.22214962900330226574105503073, 22.99490961635519343084623738707, 24.2740624544140139065224121851