L(s) = 1 | + (5.96 − 5.96i)3-s + (−8.67 − 8.67i)5-s + 1.63i·7-s − 44.1i·9-s + (−18.2 − 18.2i)11-s + (9.34 − 9.34i)13-s − 103.·15-s + 53.6·17-s + (−70.9 + 70.9i)19-s + (9.77 + 9.77i)21-s − 25.1i·23-s + 25.6i·25-s + (−102. − 102. i)27-s + (181. − 181. i)29-s + 132.·31-s + ⋯ |
L(s) = 1 | + (1.14 − 1.14i)3-s + (−0.776 − 0.776i)5-s + 0.0885i·7-s − 1.63i·9-s + (−0.498 − 0.498i)11-s + (0.199 − 0.199i)13-s − 1.78·15-s + 0.764·17-s + (−0.857 + 0.857i)19-s + (0.101 + 0.101i)21-s − 0.227i·23-s + 0.205i·25-s + (−0.729 − 0.729i)27-s + (1.15 − 1.15i)29-s + 0.768·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.915450 - 1.63843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.915450 - 1.63843i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-5.96 + 5.96i)T - 27iT^{2} \) |
| 5 | \( 1 + (8.67 + 8.67i)T + 125iT^{2} \) |
| 7 | \( 1 - 1.63iT - 343T^{2} \) |
| 11 | \( 1 + (18.2 + 18.2i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-9.34 + 9.34i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (70.9 - 70.9i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 25.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-181. + 181. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (174. + 174. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 198. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-285. - 285. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 78.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-525. - 525. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (46.5 + 46.5i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (193. - 193. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (282. - 282. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 727. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 106. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 58.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-410. + 410. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 768. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 809.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.50984743339471184568188302038, −12.03528278784069846707155941322, −10.38256285910120102256960045321, −8.801223183372475047159190039504, −8.227260077530234431606538009079, −7.47306620523798566518426335100, −5.96562827585302963912435140585, −4.08678988897818671709058270505, −2.61960046576875137581363760695, −0.889714434706940621650329653930,
2.70665095213724283747956068741, 3.72380149423249804264412804531, 4.86511250767946045767052317396, 6.93348354312631268341846971694, 8.050121014574612064720203796957, 8.980338182310754735989657577982, 10.18809299511817135484028722845, 10.77947103548330554273908616679, 12.10360516511565423776864895820, 13.54363916524704456144797701650