L(s) = 1 | + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.766 − 0.642i)5-s + (0.0348 + 0.999i)7-s + (−0.978 − 0.207i)8-s + (−0.809 + 0.587i)10-s + (0.882 + 0.469i)11-s + (−0.438 + 0.898i)13-s + (0.0348 − 0.999i)14-s + (0.961 + 0.275i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (0.848 − 0.529i)20-s + (−0.848 − 0.529i)22-s + (0.882 − 0.469i)23-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.766 − 0.642i)5-s + (0.0348 + 0.999i)7-s + (−0.978 − 0.207i)8-s + (−0.809 + 0.587i)10-s + (0.882 + 0.469i)11-s + (−0.438 + 0.898i)13-s + (0.0348 − 0.999i)14-s + (0.961 + 0.275i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (0.848 − 0.529i)20-s + (−0.848 − 0.529i)22-s + (0.882 − 0.469i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.509007201 + 0.8055401656i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.509007201 + 0.8055401656i\) |
\(L(1)\) |
\(\approx\) |
\(0.9097338858 + 0.1163222091i\) |
\(L(1)\) |
\(\approx\) |
\(0.9097338858 + 0.1163222091i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.997 - 0.0697i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.0348 + 0.999i)T \) |
| 11 | \( 1 + (0.882 + 0.469i)T \) |
| 13 | \( 1 + (-0.438 + 0.898i)T \) |
| 17 | \( 1 + (0.978 + 0.207i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.882 - 0.469i)T \) |
| 29 | \( 1 + (0.997 + 0.0697i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.719 - 0.694i)T \) |
| 43 | \( 1 + (0.997 + 0.0697i)T \) |
| 47 | \( 1 + (-0.719 + 0.694i)T \) |
| 53 | \( 1 + (0.978 + 0.207i)T \) |
| 59 | \( 1 + (0.438 - 0.898i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.978 - 0.207i)T \) |
| 79 | \( 1 + (-0.990 + 0.139i)T \) |
| 83 | \( 1 + (-0.559 + 0.829i)T \) |
| 89 | \( 1 + (-0.669 - 0.743i)T \) |
| 97 | \( 1 + (0.0348 + 0.999i)T \) |
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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
−21.554053962548415611863789697853, −21.025721591713427037240473554434, −19.9181352245405869496427170513, −19.47107079009118686491277679242, −18.5549529030368090903815697658, −17.7157854534134226347697036482, −17.03779670113043042105062052403, −16.68503587235522943507747737248, −15.3490586423928523713824476452, −14.66398573833093768738310766363, −13.86483713197451820775744700778, −12.89791698950419303672286442683, −11.69198149995202982021873553446, −10.901660806690903261782319794845, −10.1912891196373096586981407870, −9.62728953166530837632106421507, −8.582372587716251348224904297111, −7.61232213875918058886759239670, −6.831419775350887544892355826273, −6.177643789476485220525590887436, −5.04940327964169519943223660630, −3.46840699129418278826451776533, −2.72789949843018702397594126572, −1.42055941828317996470967499361, −0.61123650091439440824609253276,
1.0263037736857558896160005129, 1.8903682670174810943325336735, 2.6596193383944289210127744369, 4.164644214554986036634726607199, 5.39039976940874167370171267233, 6.232401765452310185510070290050, 7.00714362244373537892421432223, 8.29224014682846165092827141107, 8.91669063250200330225201998535, 9.55653376022332734490369903274, 10.27198034506991043638986238877, 11.46394236316870940957704239634, 12.31018863506178543167406198720, 12.66581999724526903031114773995, 14.29812412911534300325593728433, 14.77079968864856953314154144397, 15.92098832052391611030439955026, 16.72305760411241798807776297886, 17.21228136481729503472614342956, 18.02561989339751006323229409290, 18.9528295983539345424133657965, 19.4184272346053815869358175074, 20.478770029549093607744066694663, 21.33360869696242787114503873089, 21.55347057804929324246023098400