| L(s) = 1 | + (−0.681 + 0.210i)2-s + (−1.23 + 0.840i)4-s + (−2.65 − 0.400i)5-s + (−0.518 + 2.59i)7-s + (1.55 − 1.94i)8-s + (1.89 − 0.285i)10-s + (1.45 + 1.34i)11-s + (0.408 − 1.78i)13-s + (−0.191 − 1.87i)14-s + (0.442 − 1.12i)16-s + (0.176 − 2.35i)17-s + (−3.23 − 5.60i)19-s + (3.61 − 1.74i)20-s + (−1.27 − 0.612i)22-s + (−0.536 − 7.15i)23-s + ⋯ |
| L(s) = 1 | + (−0.481 + 0.148i)2-s + (−0.616 + 0.420i)4-s + (−1.18 − 0.179i)5-s + (−0.196 + 0.980i)7-s + (0.548 − 0.687i)8-s + (0.599 − 0.0903i)10-s + (0.438 + 0.406i)11-s + (0.113 − 0.495i)13-s + (−0.0512 − 0.501i)14-s + (0.110 − 0.281i)16-s + (0.0427 − 0.571i)17-s + (−0.742 − 1.28i)19-s + (0.808 − 0.389i)20-s + (−0.271 − 0.130i)22-s + (−0.111 − 1.49i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0860 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0860 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.275478 - 0.252698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.275478 - 0.252698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.518 - 2.59i)T \) |
| good | 2 | \( 1 + (0.681 - 0.210i)T + (1.65 - 1.12i)T^{2} \) |
| 5 | \( 1 + (2.65 + 0.400i)T + (4.77 + 1.47i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 1.34i)T + (0.822 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.408 + 1.78i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.176 + 2.35i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (3.23 + 5.60i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.536 + 7.15i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (2.13 - 1.02i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.98 + 5.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.06 + 0.724i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-1.15 + 1.44i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.12 - 7.67i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (4.45 - 1.37i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (-0.302 + 0.206i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-4.96 + 0.748i)T + (56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (-0.00167 - 0.00113i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (-5.23 + 9.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.12 + 3.43i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (1.23 + 0.380i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (0.184 + 0.319i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.44 + 15.1i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (4.91 - 4.56i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + 5.61T + 97T^{2} \) |
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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
−10.98025036395591587062787235469, −9.722283942375992254530884428559, −8.933399246565752785127967510305, −8.301263281419290182429408970200, −7.47739874875209483410535997594, −6.41683476008190574077469656349, −4.86516167771446964868387285774, −4.10890804894120415546605086558, −2.77146265300746240487809758526, −0.31048229209757993681129040416,
1.37321201594970829410901520950, 3.73923902849412412525462643960, 4.14292717738439332059535572958, 5.64296145441993197301270504926, 6.89386366764173461110248226821, 7.88608579104069268156804382654, 8.521725444208700507411416654754, 9.622270431064014190348864253626, 10.43876765343976878548647381010, 11.15952213082247069048765973543
