L(s) = 1 | + (−0.654 − 0.755i)3-s + (0.909 + 0.415i)7-s + (−0.142 + 0.989i)9-s + (0.281 − 0.959i)11-s + (0.415 + 0.909i)13-s + (0.540 − 0.841i)17-s + (0.540 + 0.841i)19-s + (−0.281 − 0.959i)21-s + (0.841 − 0.540i)27-s + (0.540 − 0.841i)29-s + (−0.654 + 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.142 − 0.989i)37-s + (0.415 − 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)3-s + (0.909 + 0.415i)7-s + (−0.142 + 0.989i)9-s + (0.281 − 0.959i)11-s + (0.415 + 0.909i)13-s + (0.540 − 0.841i)17-s + (0.540 + 0.841i)19-s + (−0.281 − 0.959i)21-s + (0.841 − 0.540i)27-s + (0.540 − 0.841i)29-s + (−0.654 + 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.142 − 0.989i)37-s + (0.415 − 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.522047389 - 0.2497033074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.522047389 - 0.2497033074i\) |
\(L(1)\) |
\(\approx\) |
\(1.037651693 - 0.1615392637i\) |
\(L(1)\) |
\(\approx\) |
\(1.037651693 - 0.1615392637i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.909 + 0.415i)T \) |
| 11 | \( 1 + (0.281 - 0.959i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (0.540 - 0.841i)T \) |
| 19 | \( 1 + (0.540 + 0.841i)T \) |
| 29 | \( 1 + (0.540 - 0.841i)T \) |
| 31 | \( 1 + (-0.654 + 0.755i)T \) |
| 37 | \( 1 + (0.142 - 0.989i)T \) |
| 41 | \( 1 + (0.142 + 0.989i)T \) |
| 43 | \( 1 + (-0.654 - 0.755i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.909 + 0.415i)T \) |
| 61 | \( 1 + (0.755 + 0.654i)T \) |
| 67 | \( 1 + (-0.959 + 0.281i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.540 + 0.841i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (0.142 - 0.989i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.989 + 0.142i)T \) |
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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
−20.31460663038695296464801581002, −19.75864673862949372338395477303, −18.33382831647523029533254318712, −17.92700246753743442071153636522, −17.16951686394015479508632376878, −16.726449688070301908722485467602, −15.62789587593584075128604527758, −15.10921125806823309988297257112, −14.53052856228195479438921064546, −13.533843771118023699740898118408, −12.57851146936959337720834552112, −11.90664615898791873525394793245, −11.078931609391989721530803271210, −10.51154331346045383474692382490, −9.84186027993739052135186896217, −8.945666907140556372135401694896, −8.05072979762669337853724364982, −7.217405850013236689549893369154, −6.3123153438539767493632806782, −5.3408392505973452840428122179, −4.81515761010944166736519990399, −3.96553013045215017251211131453, −3.16320803916852924760411351239, −1.76430787103335278230712494938, −0.81577437954677447939253589142,
0.92405520487875700272908203272, 1.64366215365850585223129058523, 2.62178242178594881225615717774, 3.78074451229325167445188164428, 4.85789192170924945098343639797, 5.58408979498914559969402256337, 6.22549706720153659086167396040, 7.14705185035097308230863531195, 7.93899840594883632304689320742, 8.58866726568786680284027377938, 9.50615089579738406603367090163, 10.63624966337065196099685008981, 11.38413966350269623401228906647, 11.79134766212403074968849064118, 12.46501633494981817063039561371, 13.6209828550812669727225601195, 14.02381467039371566875455681978, 14.722960270639168014418427066389, 16.10197903603651273301897884325, 16.34696286132427547664683390353, 17.246433800416089065429292795189, 18.07013205080282503830610335790, 18.54243635860243687978177791068, 19.12002644115713839799722588737, 19.9893345793645368648206918639