L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.222 + 0.974i)5-s + (0.222 − 0.974i)8-s + (0.733 − 0.680i)10-s + (0.900 − 0.433i)11-s + (−0.826 − 0.563i)13-s + (−0.733 + 0.680i)16-s + (0.365 − 0.930i)17-s + (0.5 − 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.365 + 0.930i)26-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.222 + 0.974i)5-s + (0.222 − 0.974i)8-s + (0.733 − 0.680i)10-s + (0.900 − 0.433i)11-s + (−0.826 − 0.563i)13-s + (−0.733 + 0.680i)16-s + (0.365 − 0.930i)17-s + (0.5 − 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.365 + 0.930i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8112466471 - 0.2534044556i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8112466471 - 0.2534044556i\) |
\(L(1)\) |
\(\approx\) |
\(0.7377406261 - 0.1216889453i\) |
\(L(1)\) |
\(\approx\) |
\(0.7377406261 - 0.1216889453i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.826 - 0.563i)T \) |
| 5 | \( 1 + (-0.222 + 0.974i)T \) |
| 11 | \( 1 + (0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.826 - 0.563i)T \) |
| 17 | \( 1 + (0.365 - 0.930i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.988 - 0.149i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.988 + 0.149i)T \) |
| 41 | \( 1 + (0.955 - 0.294i)T \) |
| 43 | \( 1 + (0.955 + 0.294i)T \) |
| 47 | \( 1 + (0.826 + 0.563i)T \) |
| 53 | \( 1 + (0.988 + 0.149i)T \) |
| 59 | \( 1 + (0.955 + 0.294i)T \) |
| 61 | \( 1 + (-0.365 + 0.930i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.0747 - 0.997i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.826 - 0.563i)T \) |
| 89 | \( 1 + (0.826 - 0.563i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−24.442942726788054674289067808970, −23.5506594090285986989484445728, −22.67092583258916913385174126304, −21.41486635316669681797392948434, −20.45028697710090110880031085927, −19.64222749086849024868439774133, −19.13862404524254965493567513815, −17.865243530698717960696662818455, −17.11885042013855938405312584214, −16.498974407207087856944214054197, −15.70742044542336669036023993226, −14.60002896653433994877706907878, −13.99222501829436276380020668372, −12.32372080230205444366189716405, −11.99111740445398162656557441226, −10.507279219312227253349389805371, −9.6882655084278459659882196995, −8.83056660390952910478323557998, −8.067143548890563873519772103143, −7.0591287461109681298820733642, −6.053637303838026730063975464, −4.98233099708697555887816916360, −3.99643779731975713405731540512, −2.07026893878881387249922029820, −1.06285790282199492272017649919,
0.81332968671421835074987283098, 2.44523917040251896419864215558, 3.145677113144361060080349929917, 4.274855801944762828468894852837, 5.932680516634665478297589402258, 7.14117598929869053913781254764, 7.62469595781329805531727826794, 8.91425161989566257730529707491, 9.79037642553492400899637028738, 10.56336002897283208792767118103, 11.66730825789346243328512636426, 11.96489076877496058337474147540, 13.448268890971837853433254051320, 14.31702403103929781051279798694, 15.46422219256460541637338882454, 16.21882387577317267308024915454, 17.49564375037653120911506744355, 17.81611456910409732564877668585, 19.048020690657769954611867185943, 19.44463561430619581486148763265, 20.32207193066252761740563224828, 21.41822943322396758650423221622, 22.24805606149515502792494690844, 22.699048310577710049995248288450, 24.18386352063063838166340669754