| L(s) = 1 | + (−0.439 + 2.49i)2-s + (−4.14 − 1.50i)4-s + (−0.358 + 0.300i)5-s + (3.03 − 1.10i)7-s + (3.05 − 5.28i)8-s + (−0.592 − 1.02i)10-s + (2.37 + 1.99i)11-s + (−0.379 − 2.15i)13-s + (1.41 + 8.04i)14-s + (5.08 + 4.26i)16-s + (1.5 + 2.59i)17-s + (−0.0209 + 0.0362i)19-s + (1.93 − 0.705i)20-s + (−6.02 + 5.05i)22-s + (5.73 + 2.08i)23-s + ⋯ |
| L(s) = 1 | + (−0.310 + 1.76i)2-s + (−2.07 − 0.754i)4-s + (−0.160 + 0.134i)5-s + (1.14 − 0.417i)7-s + (1.07 − 1.86i)8-s + (−0.187 − 0.324i)10-s + (0.717 + 0.601i)11-s + (−0.105 − 0.596i)13-s + (0.379 + 2.15i)14-s + (1.27 + 1.06i)16-s + (0.363 + 0.630i)17-s + (−0.00480 + 0.00832i)19-s + (0.433 − 0.157i)20-s + (−1.28 + 1.07i)22-s + (1.19 + 0.435i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.366011 + 1.22256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.366011 + 1.22256i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.439 - 2.49i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.358 - 0.300i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.03 + 1.10i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.37 - 1.99i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.379 + 2.15i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0209 - 0.0362i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.73 - 2.08i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.14 - 6.47i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.85 - 2.12i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.79 + 3.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.33 + 7.58i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.450 + 0.378i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (9.07 - 3.30i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 + (-6.53 + 5.48i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 0.433i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.73 - 9.85i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.91 - 10.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.11 + 7.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.91 + 10.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.262 + 1.48i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (7.93 - 13.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.2 - 11.9i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.53529802184576089599559980535, −9.544162304236430462267243599961, −8.713208173875048968228613029030, −7.962419012127302454940843398799, −7.27810161903010871777505816521, −6.62874843009933123106824394541, −5.38271481210261889698086677718, −4.85536853846760588303286308595, −3.68136969772610726992969723728, −1.32792992481616282169851449775,
0.911335230770448958832362849511, 2.07359958268742720250439342234, 3.14360952021820085942727265727, 4.34748643567407616672423956551, 4.97162961189259660385637688532, 6.47860926988457701863563141083, 8.031434216769767004403874556081, 8.536455623366138002564135392945, 9.403120570028869218356748571413, 10.09967957898776803023951041496
