| L(s) = 1 | + (0.642 − 0.766i)5-s + (−0.642 − 0.766i)11-s + (0.342 + 0.939i)13-s + 17-s − i·19-s + (0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s + (0.173 − 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.173 − 0.984i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯ |
| L(s) = 1 | + (0.642 − 0.766i)5-s + (−0.642 − 0.766i)11-s + (0.342 + 0.939i)13-s + 17-s − i·19-s + (0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s + (0.173 − 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.173 − 0.984i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.237920768 - 1.386787602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237920768 - 1.386787602i\) |
| \(L(1)\) |
\(\approx\) |
\(1.225246623 - 0.2229405475i\) |
| \(L(1)\) |
\(\approx\) |
\(1.225246623 - 0.2229405475i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (0.642 - 0.766i)T \) |
| 11 | \( 1 + (-0.642 - 0.766i)T \) |
| 13 | \( 1 + (0.342 + 0.939i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.342 + 0.939i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.984 - 0.173i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.342 + 0.939i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.173 - 0.984i)T \) |
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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
−18.93860708157127437834997514742, −18.20748183252897118578083820782, −17.60805232068833890460572801850, −17.21351516040989812134028942429, −16.08314275372702149350977087358, −15.386531881405292050780182401998, −14.8740126813335150711311824382, −14.12373455719031256212516972258, −13.26625669957952465892670333415, −12.890441285316079832671300353674, −11.916847476286841376553607556869, −11.03744725124317117186303761975, −10.43302189122641473985051677041, −9.876062835012705577989840278627, −9.11229846535643974481320314506, −8.142962092352390215986357233876, −7.31922495073978643323916725766, −6.86154053643920025919042899255, −5.71981104642263177984178118352, −5.36752540462612974884659969620, −4.34400609191867953349917878784, −3.14759753158771677859120136915, −2.78710891243602503977938345172, −1.76175555911759449559801704035, −0.7799184831680491982718127294,
0.51060383408546252802199072687, 1.372706877071513127742405510709, 2.14042096513908824552215835301, 3.224362527288642543705135798707, 3.98097115562460538861712352316, 5.06394218568530271187364033126, 5.51348794811992302986095333382, 6.30618004662489356946221539271, 7.15927324448780043708385898999, 8.250265526936211529119151841421, 8.58109832603241610385870302120, 9.53139908235312856687140556030, 10.0856017018286353132477032621, 10.9504065476709657934515875983, 11.70502360127458845694208013335, 12.5472789477020733084833945116, 13.07531381351691089999923830086, 13.9003595551358993586654393211, 14.32423481173522647414468275352, 15.30112857744290787303946676616, 16.242109260907344464973120433819, 16.66068681173114250622512187217, 17.09011157823907049739661719475, 18.27027713694113619040908150064, 18.63802982035615423453248391128
