L(s) = 1 | + (0.286 + 0.957i)5-s + (0.396 + 0.918i)7-s + (−0.973 + 0.230i)11-s + (0.835 + 0.549i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)19-s + (0.396 − 0.918i)23-s + (−0.835 + 0.549i)25-s + (0.0581 + 0.998i)29-s + (−0.993 + 0.116i)31-s + (−0.766 + 0.642i)35-s + (−0.766 − 0.642i)37-s + (0.893 − 0.448i)41-s + (0.686 + 0.727i)43-s + (−0.993 − 0.116i)47-s + ⋯ |
L(s) = 1 | + (0.286 + 0.957i)5-s + (0.396 + 0.918i)7-s + (−0.973 + 0.230i)11-s + (0.835 + 0.549i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)19-s + (0.396 − 0.918i)23-s + (−0.835 + 0.549i)25-s + (0.0581 + 0.998i)29-s + (−0.993 + 0.116i)31-s + (−0.766 + 0.642i)35-s + (−0.766 − 0.642i)37-s + (0.893 − 0.448i)41-s + (0.686 + 0.727i)43-s + (−0.993 − 0.116i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6878621206 + 1.091367552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6878621206 + 1.091367552i\) |
\(L(1)\) |
\(\approx\) |
\(0.9821552129 + 0.4413526424i\) |
\(L(1)\) |
\(\approx\) |
\(0.9821552129 + 0.4413526424i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.286 + 0.957i)T \) |
| 7 | \( 1 + (0.396 + 0.918i)T \) |
| 11 | \( 1 + (-0.973 + 0.230i)T \) |
| 13 | \( 1 + (0.835 + 0.549i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.396 - 0.918i)T \) |
| 29 | \( 1 + (0.0581 + 0.998i)T \) |
| 31 | \( 1 + (-0.993 + 0.116i)T \) |
| 37 | \( 1 + (-0.766 - 0.642i)T \) |
| 41 | \( 1 + (0.893 - 0.448i)T \) |
| 43 | \( 1 + (0.686 + 0.727i)T \) |
| 47 | \( 1 + (-0.993 - 0.116i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.973 - 0.230i)T \) |
| 61 | \( 1 + (-0.597 + 0.802i)T \) |
| 67 | \( 1 + (0.0581 - 0.998i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (0.893 + 0.448i)T \) |
| 83 | \( 1 + (-0.893 - 0.448i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (-0.286 + 0.957i)T \) |
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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
−22.75883176956609461693961307177, −21.604131252377642475163132079450, −20.77740767413796992659459647428, −20.36554494735952499088427607098, −19.54887670723388924225285158212, −18.24933820560242127391027235752, −17.67830662083596879961205720543, −16.87296819581070784240928810878, −15.92189509085608260774073988575, −15.42161578514150254190164550029, −13.88159455135480035255188062700, −13.46995284205791596213046005851, −12.794456921815297443379192145193, −11.502475011291340812140487745264, −10.83595708755746316785891269359, −9.8335360130116229435457749083, −8.91997216132461684605619818758, −7.9996330581976922874889590443, −7.27833379360689106453962791087, −5.90265082613209261043091172469, −5.11545352199020578812429630677, −4.24683246462142008092692610742, −3.08883310707005897840838576012, −1.69594637574095347851786787690, −0.63224927985001193751355219635,
1.7417028877093498989375108211, 2.55601341315147645341658872714, 3.56790814425993800871083992701, 4.91125111600411317987759707747, 5.8023026403198880270786388414, 6.677712027879753036828321630818, 7.641211780518720994515801883924, 8.67825613137354493159706104355, 9.46663408098212354728353452750, 10.7562561626514088814537862369, 11.03586026025266060308704475187, 12.22955047779384488785905127729, 13.1102288439480320797535043040, 14.10401700776690679355360896054, 14.78716102179923616666848692143, 15.64122245526097625800516082841, 16.32293172401475273065865581305, 17.78611690222731898813585467317, 18.19328137757536185611601268289, 18.74570682867736816358335732532, 19.809192407174829682157029017131, 20.97386161056314538195569036666, 21.39226705188442524532464476733, 22.38062664364671057986250581923, 22.93719118629834419683145316361