| L(s) = 1 | + (0.152 − 0.866i)2-s + (1.15 + 0.419i)4-s + (−2.97 + 2.49i)5-s + (2.05 − 0.747i)7-s + (1.41 − 2.45i)8-s + (1.70 + 2.95i)10-s + (−0.124 − 0.104i)11-s + (0.418 + 2.37i)13-s + (−0.333 − 1.89i)14-s + (−0.0320 − 0.0269i)16-s + (1.5 + 2.59i)17-s + (−1.79 + 3.11i)19-s + (−4.47 + 1.62i)20-s + (−0.109 + 0.0918i)22-s + (2.66 + 0.970i)23-s + ⋯ |
| L(s) = 1 | + (0.107 − 0.612i)2-s + (0.576 + 0.209i)4-s + (−1.32 + 1.11i)5-s + (0.775 − 0.282i)7-s + (0.501 − 0.868i)8-s + (0.539 + 0.934i)10-s + (−0.0375 − 0.0314i)11-s + (0.116 + 0.658i)13-s + (−0.0891 − 0.505i)14-s + (−0.00802 − 0.00673i)16-s + (0.363 + 0.630i)17-s + (−0.412 + 0.714i)19-s + (−0.999 + 0.363i)20-s + (−0.0233 + 0.0195i)22-s + (0.555 + 0.202i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59841 + 0.378831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59841 + 0.378831i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.152 + 0.866i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (2.97 - 2.49i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.05 + 0.747i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.124 + 0.104i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.418 - 2.37i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.79 - 3.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.66 - 0.970i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.16 - 6.61i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.87 - 1.77i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.31 - 5.74i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.00 - 5.71i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.76 + 4.00i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.95 + 2.52i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + (-3.92 + 3.29i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.55 + 1.29i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.01 + 5.77i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (7.65 + 13.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.34 - 7.51i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.220 - 1.24i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.47 + 8.34i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.86 + 6.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.99 + 2.51i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.74076082966637133350185446152, −10.06300176473016586929549959890, −8.489044542261792879982963327095, −7.77744913862373799683163731793, −7.07473656269956973493124498625, −6.32326366651741578901292062801, −4.59950112280656088816638426058, −3.72944367143785401910024213266, −2.97578365041511978315825383235, −1.56299403447934425558187594264,
0.896785076849626519119223588539, 2.57161396742161245230964763244, 4.17183283170238557999232869913, 4.97531739829645053589892108330, 5.71588557827259573257513305231, 7.03086969152377841503946826539, 7.83555809848841201907694030906, 8.268502682218169675782186226414, 9.175686480440900708216022939801, 10.50602452707098231200793315984
