| L(s) = 1 | + (−2.26 − 0.630i)2-s + (3.02 + 1.82i)4-s + (2.74 − 0.106i)5-s + (2.01 + 0.314i)7-s + (−2.47 − 2.62i)8-s + (−6.28 − 1.48i)10-s + (−0.475 − 3.48i)11-s + (3.06 − 4.47i)13-s + (−4.36 − 1.98i)14-s + (0.657 + 1.24i)16-s + (−6.06 + 3.04i)17-s + (−0.238 − 4.10i)19-s + (8.49 + 4.68i)20-s + (−1.11 + 8.19i)22-s + (3.06 − 7.92i)23-s + ⋯ |
| L(s) = 1 | + (−1.60 − 0.446i)2-s + (1.51 + 0.912i)4-s + (1.22 − 0.0476i)5-s + (0.760 + 0.118i)7-s + (−0.874 − 0.926i)8-s + (−1.98 − 0.471i)10-s + (−0.143 − 1.05i)11-s + (0.851 − 1.24i)13-s + (−1.16 − 0.529i)14-s + (0.164 + 0.312i)16-s + (−1.47 + 0.739i)17-s + (−0.0548 − 0.941i)19-s + (1.89 + 1.04i)20-s + (−0.238 + 1.74i)22-s + (0.638 − 1.65i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.690620 - 0.587587i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.690620 - 0.587587i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.26 + 0.630i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-2.74 + 0.106i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-2.01 - 0.314i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.475 + 3.48i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-3.06 + 4.47i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (6.06 - 3.04i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.238 + 4.10i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-3.06 + 7.92i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-3.45 - 2.46i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (3.03 - 3.48i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (1.71 + 2.30i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (0.270 - 1.05i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-2.62 - 2.38i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-1.07 - 1.23i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (0.605 - 3.43i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.85 + 2.39i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (1.49 - 0.902i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (3.44 - 2.46i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (3.29 + 11.0i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (-4.93 + 1.17i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-2.80 + 2.75i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-3.48 - 13.5i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-1.52 + 5.08i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (8.20 + 0.318i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.37077327634634420007350304211, −9.131497259940675969578536100990, −8.663667005230481736087415516674, −8.149613300385877589415022413359, −6.81101174624532242782630912039, −6.00121593960826208019010579244, −4.86899675239945906361634851884, −2.98318490405130809084463901776, −2.03454454687944953940800331827, −0.831870114333679526280150329435,
1.56597672550597917509399440179, 2.09365863876717194150960424216, 4.30164531038385963313239298916, 5.54028219605313381809004778967, 6.55897268198204042888137832557, 7.16522767433546689291871459280, 8.128040006753443697372210617156, 9.119102271366149758700519604532, 9.462103286994288440500355173478, 10.26811556203966752469446287567
