| 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
−10.26811556203966752469446287567, −9.462103286994288440500355173478, −9.119102271366149758700519604532, −8.128040006753443697372210617156, −7.16522767433546689291871459280, −6.55897268198204042888137832557, −5.54028219605313381809004778967, −4.30164531038385963313239298916, −2.09365863876717194150960424216, −1.56597672550597917509399440179,
0.831870114333679526280150329435, 2.03454454687944953940800331827, 2.98318490405130809084463901776, 4.86899675239945906361634851884, 6.00121593960826208019010579244, 6.81101174624532242782630912039, 8.149613300385877589415022413359, 8.663667005230481736087415516674, 9.131497259940675969578536100990, 10.37077327634634420007350304211
