L(s) = 1 | + (0.134 − 0.990i)2-s + (−0.393 + 0.919i)3-s + (−0.963 − 0.266i)4-s + (−0.995 − 0.0896i)5-s + (0.858 + 0.512i)6-s + (−0.393 + 0.919i)8-s + (−0.691 − 0.722i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (0.134 − 0.990i)13-s + (0.473 − 0.880i)15-s + (0.858 + 0.512i)16-s + (−0.550 − 0.834i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.936 + 0.351i)20-s + ⋯ |
L(s) = 1 | + (0.134 − 0.990i)2-s + (−0.393 + 0.919i)3-s + (−0.963 − 0.266i)4-s + (−0.995 − 0.0896i)5-s + (0.858 + 0.512i)6-s + (−0.393 + 0.919i)8-s + (−0.691 − 0.722i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (0.134 − 0.990i)13-s + (0.473 − 0.880i)15-s + (0.858 + 0.512i)16-s + (−0.550 − 0.834i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.936 + 0.351i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5707111401 + 0.1608743622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5707111401 + 0.1608743622i\) |
\(L(1)\) |
\(\approx\) |
\(0.6592981393 - 0.1204734736i\) |
\(L(1)\) |
\(\approx\) |
\(0.6592981393 - 0.1204734736i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.134 - 0.990i)T \) |
| 3 | \( 1 + (-0.393 + 0.919i)T \) |
| 5 | \( 1 + (-0.995 - 0.0896i)T \) |
| 13 | \( 1 + (0.134 - 0.990i)T \) |
| 17 | \( 1 + (-0.550 - 0.834i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (-0.0448 + 0.998i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.0448 + 0.998i)T \) |
| 41 | \( 1 + (-0.393 + 0.919i)T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.473 + 0.880i)T \) |
| 53 | \( 1 + (-0.550 + 0.834i)T \) |
| 59 | \( 1 + (-0.393 - 0.919i)T \) |
| 61 | \( 1 + (-0.550 - 0.834i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.936 - 0.351i)T \) |
| 73 | \( 1 + (0.473 - 0.880i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.134 + 0.990i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−23.44958429063665239058396355590, −22.897723135571585684504114656979, −22.10680505088229394472351343516, −20.96556696956714918915956980355, −19.57077679563009891424878496892, −18.92396932634511801463707759881, −18.4126718921088589441433362039, −17.1057466705457098585892980930, −16.819591237732229104800077251086, −15.71573384843219571726301707644, −14.89719349263809807108154134690, −14.0447138123891096242450965677, −13.07739642241547671947979607597, −12.35902719117933183042406512893, −11.50869864968367488746962914772, −10.45938727578810018788814907990, −8.83805835076351191265429555894, −8.28806657663919527183659448953, −7.32382693420669513271089275325, −6.64485462021222412033401403212, −5.83903441682742633429456445498, −4.54999424762421109629028777662, −3.79285323685632670202724164035, −2.13046970445572741504993516177, −0.41654479396899573287860411870,
0.96336385675873517021718399022, 2.90611775218965597664909202581, 3.46657530413988145159407617137, 4.698368921652462070585991426995, 5.04562916955608824046250483737, 6.52211422599893945624346678670, 8.028064221421906354950383270493, 8.898676004941408761554569913067, 9.72413570057245569219277876505, 10.95988925778592886332447620860, 11.07167047532896737152599266508, 12.17959473348963900022319858366, 12.91775287068481453214039971420, 14.109531168460296392698562869528, 15.19687390070897894403565501134, 15.614775958151332710397616153934, 16.8138830760289428084365103499, 17.68656221669356197248550214610, 18.53869724612404923968920752833, 19.73002394837044945213853378030, 20.09714252051277655783106617694, 20.96901266949235447466459175653, 21.82652644408896000071598274864, 22.55321650355674294804473808837, 23.247816937180489448585201354872