| L(s) = 1 | + (−0.367 + 2.08i)2-s + (−2.33 − 0.848i)4-s + (−2.05 + 1.72i)5-s + (−0.913 + 0.332i)7-s + (0.508 − 0.880i)8-s + (−2.83 − 4.91i)10-s + (−0.242 − 0.203i)11-s + (−0.262 − 1.49i)13-s + (−0.357 − 2.02i)14-s + (−2.15 − 1.80i)16-s + (−0.587 − 1.01i)17-s + (−3.11 + 5.38i)19-s + (6.25 − 2.27i)20-s + (0.513 − 0.431i)22-s + (−2.03 − 0.739i)23-s + ⋯ |
| L(s) = 1 | + (−0.259 + 1.47i)2-s + (−1.16 − 0.424i)4-s + (−0.919 + 0.771i)5-s + (−0.345 + 0.125i)7-s + (0.179 − 0.311i)8-s + (−0.897 − 1.55i)10-s + (−0.0731 − 0.0614i)11-s + (−0.0729 − 0.413i)13-s + (−0.0955 − 0.541i)14-s + (−0.538 − 0.451i)16-s + (−0.142 − 0.246i)17-s + (−0.713 + 1.23i)19-s + (1.39 − 0.508i)20-s + (0.109 − 0.0919i)22-s + (−0.423 − 0.154i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.201395 - 0.533364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.201395 - 0.533364i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.367 - 2.08i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (2.05 - 1.72i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.913 - 0.332i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.242 + 0.203i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.262 + 1.49i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.587 + 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 - 5.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.03 + 0.739i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.764 - 4.33i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.15 - 2.96i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.01 - 5.75i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.28 - 3.59i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.32 + 0.846i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + (1.32 - 1.10i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.953 - 0.347i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.148 - 0.843i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.79 - 8.31i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.94 - 11.0i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.813 + 4.61i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.74 + 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.25 + 3.56i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.76067591283428774449171075105, −11.73524342284672685383379827402, −10.68724634082140566231587053030, −9.587488536502653609698786402786, −8.298736337995241359672143420724, −7.78308810814326307570128622485, −6.73082346076951154874627748014, −6.00320039024274317983384811772, −4.57796455629676751403243457439, −3.08734923579589338394855229468,
0.48864303727120602794688119658, 2.35191046409127173590920588609, 3.84588573071976402806027242934, 4.61527443820146802015481353775, 6.48311134329522108664516224533, 7.924184911862643568685152635706, 8.900337562483359920622639214954, 9.681439511088860813832867265042, 10.77353372407261877018044048385, 11.53970081333930256140843061836
