| L(s) = 1 | + (−1.50 − 1.26i)2-s + (0.326 + 1.85i)4-s + (−3.47 − 1.26i)5-s + (−0.407 + 2.31i)7-s + (−0.118 + 0.205i)8-s + (3.64 + 6.31i)10-s + (2.04 − 0.745i)11-s + (−3.61 + 3.03i)13-s + (3.54 − 2.97i)14-s + (3.97 − 1.44i)16-s + (1.46 + 2.54i)17-s + (3.11 − 5.39i)19-s + (1.20 − 6.85i)20-s + (−4.03 − 1.46i)22-s + (0.0901 + 0.511i)23-s + ⋯ |
| L(s) = 1 | + (−1.06 − 0.895i)2-s + (0.163 + 0.925i)4-s + (−1.55 − 0.566i)5-s + (−0.154 + 0.873i)7-s + (−0.0419 + 0.0727i)8-s + (1.15 + 1.99i)10-s + (0.617 − 0.224i)11-s + (−1.00 + 0.840i)13-s + (0.946 − 0.794i)14-s + (0.992 − 0.361i)16-s + (0.355 + 0.616i)17-s + (0.714 − 1.23i)19-s + (0.270 − 1.53i)20-s + (−0.859 − 0.312i)22-s + (0.0187 + 0.106i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.367425 - 0.326980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.367425 - 0.326980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (1.50 + 1.26i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (3.47 + 1.26i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.407 - 2.31i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.04 + 0.745i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.61 - 3.03i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.46 - 2.54i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.11 + 5.39i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0901 - 0.511i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.67 + 2.24i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.747 + 4.23i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.20 + 2.08i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.91 + 1.60i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1 - 0.363i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0412 - 0.233i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 4.66T + 53T^{2} \) |
| 59 | \( 1 + (-12.5 - 4.55i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.638 - 3.61i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 9.19i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.601 + 1.04i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.34 + 4.05i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.80 + 8.22i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.65 + 7.26i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.349 - 0.605i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.65 + 2.42i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.08237816345623067091421844490, −9.108858999396439335856959573999, −8.875383560979489905080447657147, −7.86931019622115137161541595672, −7.13585524901299626439431500014, −5.62234669950247200055937024844, −4.46629065031700283669698593334, −3.39014910536183546339529289560, −2.17268295707162211961773773007, −0.59359622012803051811145355827,
0.77307404679102834944936487064, 3.25451310757297421444214225362, 3.98325233066284285627191457485, 5.40069490714374318167293787588, 6.86359321132378301030494651638, 7.23135401094740291530351107998, 7.83731234357420520229996884766, 8.556698400786951023045955687123, 9.827802615794803180327509703354, 10.20521597494256122202113624093
