| L(s) = 1 | + (−0.183 + 1.03i)2-s + (0.834 + 0.303i)4-s + (1.33 − 1.12i)5-s + (−2.31 + 0.841i)7-s + (−1.52 + 2.63i)8-s + (0.920 + 1.59i)10-s + (0.960 + 0.806i)11-s + (−0.789 − 4.47i)13-s + (−0.450 − 2.55i)14-s + (−1.09 − 0.921i)16-s + (−3.32 − 5.75i)17-s + (−0.124 + 0.215i)19-s + (1.45 − 0.530i)20-s + (−1.01 + 0.849i)22-s + (0.791 + 0.287i)23-s + ⋯ |
| L(s) = 1 | + (−0.129 + 0.734i)2-s + (0.417 + 0.151i)4-s + (0.598 − 0.501i)5-s + (−0.873 + 0.317i)7-s + (−0.538 + 0.932i)8-s + (0.291 + 0.504i)10-s + (0.289 + 0.243i)11-s + (−0.219 − 1.24i)13-s + (−0.120 − 0.682i)14-s + (−0.274 − 0.230i)16-s + (−0.806 − 1.39i)17-s + (−0.0285 + 0.0495i)19-s + (0.325 − 0.118i)20-s + (−0.215 + 0.181i)22-s + (0.164 + 0.0600i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.893377 + 0.442838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.893377 + 0.442838i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.183 - 1.03i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.33 + 1.12i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.31 - 0.841i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.960 - 0.806i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.789 + 4.47i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.32 + 5.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.124 - 0.215i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.791 - 0.287i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0889 + 0.504i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.770 - 0.280i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.41 - 8.02i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.31 - 2.78i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (4.98 - 1.81i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + (-2.30 + 1.93i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.70 - 0.986i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 9.93i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.0447 - 0.0774i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.66 + 4.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.829 + 4.70i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.39 - 7.91i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.35 + 5.80i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.20 - 3.52i)T + (16.8 + 95.5i)T^{2} \) |
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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
−14.79165695925655715215979779928, −13.41699920201409352569483764496, −12.53518288694509463104389523787, −11.34582358952350661749352936068, −9.821708437527237203557539705835, −8.866165455814851765675026646893, −7.48471077020524467910435200652, −6.33634652812503533449301928274, −5.23343862859717802680960239048, −2.78558948457901225601709292024,
2.16068297297437154876442698701, 3.80003426340643955048360263003, 6.22642883041752329690605616524, 6.86109191877395207907403819444, 8.978463299224768105602215963313, 10.04169862870961825985496177880, 10.79623880000335488468427631511, 11.92977543063548179942743999945, 13.01480996156960198889011979614, 14.08129204109670002615835982899
