| L(s) = 1 | + 2.44i·5-s + (−1 + 2.44i)7-s − 2.44i·11-s − 4.89i·13-s − 2.44i·17-s + 2·19-s + 7.34i·23-s − 0.999·25-s + 6·29-s + 8·31-s + (−5.99 − 2.44i)35-s − 4·37-s + 7.34i·41-s − 4.89i·43-s + 12·47-s + ⋯ |
| L(s) = 1 | + 1.09i·5-s + (−0.377 + 0.925i)7-s − 0.738i·11-s − 1.35i·13-s − 0.594i·17-s + 0.458·19-s + 1.53i·23-s − 0.199·25-s + 1.11·29-s + 1.43·31-s + (−1.01 − 0.414i)35-s − 0.657·37-s + 1.14i·41-s − 0.747i·43-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.769934436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.769934436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
| good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 2.44iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 7.34iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 + 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 9.79iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−8.469204947544295420021501899176, −7.900989008141513844519779175484, −7.04553103045493731933468230575, −6.37479369745579898372395713839, −5.63001794155721557041355695650, −5.10885848877651099318950978038, −3.68441074831981262996201358204, −2.95363662569522248918609961315, −2.59502862412262800116140544329, −0.947852183210974502938628026102,
0.65312076414784849846568338763, 1.59666040841639956781098124158, 2.71239194733128106647376662670, 4.05633313539108606858140977856, 4.40567675297270199358108154002, 5.08551732690939680305511450819, 6.28730855364845039368042757494, 6.79004559055561789030135910470, 7.53688459541587718216138216824, 8.503764164737708725993132932199
