L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.406 + 0.913i)5-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)8-s + (0.5 + 0.866i)10-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.104 + 0.994i)17-s + (−0.743 + 0.669i)19-s + (0.994 + 0.104i)20-s + (0.5 − 0.866i)23-s + (−0.669 − 0.743i)25-s + (−0.406 + 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.406 + 0.913i)5-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)8-s + (0.5 + 0.866i)10-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.104 + 0.994i)17-s + (−0.743 + 0.669i)19-s + (0.994 + 0.104i)20-s + (0.5 − 0.866i)23-s + (−0.669 − 0.743i)25-s + (−0.406 + 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0366 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0366 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.063764594 - 1.025482060i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.063764594 - 1.025482060i\) |
\(L(1)\) |
\(\approx\) |
\(0.9502654786 - 0.4071766053i\) |
\(L(1)\) |
\(\approx\) |
\(0.9502654786 - 0.4071766053i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (-0.406 + 0.913i)T \) |
| 7 | \( 1 + (-0.743 - 0.669i)T \) |
| 17 | \( 1 + (0.104 + 0.994i)T \) |
| 19 | \( 1 + (-0.743 + 0.669i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.994 - 0.104i)T \) |
| 37 | \( 1 + (-0.743 - 0.669i)T \) |
| 41 | \( 1 + (-0.743 + 0.669i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.743 - 0.669i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.951 - 0.309i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.994 - 0.104i)T \) |
| 73 | \( 1 + (0.951 - 0.309i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.994 - 0.104i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.994 - 0.104i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.04710988084550094892699195577, −20.48009214877878633206223804017, −19.40864856683111098834872514355, −18.77218680833254729377791411091, −17.66349034793124055703575904912, −17.00735553382127953341340422650, −16.267353723530359151050227098600, −15.50603770496211721410503760760, −15.2782534483585968669714195168, −13.95091413763414294868942371930, −13.33297882908363245481388924582, −12.56875964914179825751545115033, −12.02878759364322144840101879354, −11.19641462300051327132527230553, −9.654330385087196701219002001904, −9.032748182891069457166422098344, −8.39933632819120753221130516911, −7.38928622053941031624764750032, −6.68740268919318117596211288039, −5.604797256044205070025019230734, −5.10672333908411469847029883119, −4.12538462977804092993227116728, −3.28099386543143623270490776963, −2.2632892526807249357298768128, −0.56817379290075541953317546363,
0.39920937718230446024127461757, 1.694723177369919150289678110115, 2.71701100545890637563501054971, 3.63527737855006101553701370667, 4.00789436798076226169546754741, 5.236468776351814106668894110296, 6.38340091152586351233707556930, 6.75128837464135638140199207561, 7.96194348035261437979034093726, 9.02544149212411630373925139940, 10.07467976009511116414543574809, 10.6160466351617344603182304000, 11.08485808111278840425548552490, 12.28966990358554278660318599217, 12.74589863281441853324646519746, 13.625252555526322305281295991399, 14.56806848972274277786049134481, 14.85022524639297536863221032051, 15.92700669313030732892492981083, 16.74496219470515478060467279765, 17.83442068113604506126766516859, 18.65947443556026179102877427143, 19.285414056703468993399100425649, 19.76639939217088340372616298929, 20.63142504352797349115206856353