| L(s) = 1 | + (0.524 − 0.439i)2-s + (−0.266 + 1.50i)4-s + (−0.984 + 0.358i)5-s + (−0.0209 − 0.118i)7-s + (1.20 + 2.09i)8-s + (−0.358 + 0.620i)10-s + (5.10 + 1.85i)11-s + (−3.50 − 2.94i)13-s + (−0.0632 − 0.0530i)14-s + (−1.32 − 0.482i)16-s + (−2.38 + 4.13i)17-s + (0.294 + 0.509i)19-s + (−0.278 − 1.58i)20-s + (3.49 − 1.27i)22-s + (−1.35 + 7.67i)23-s + ⋯ |
| L(s) = 1 | + (0.370 − 0.310i)2-s + (−0.133 + 0.754i)4-s + (−0.440 + 0.160i)5-s + (−0.00791 − 0.0448i)7-s + (0.427 + 0.739i)8-s + (−0.113 + 0.196i)10-s + (1.53 + 0.560i)11-s + (−0.971 − 0.815i)13-s + (−0.0168 − 0.0141i)14-s + (−0.331 − 0.120i)16-s + (−0.579 + 1.00i)17-s + (0.0675 + 0.116i)19-s + (−0.0623 − 0.353i)20-s + (0.744 − 0.271i)22-s + (−0.282 + 1.60i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09484 + 0.974328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09484 + 0.974328i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.524 + 0.439i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (0.984 - 0.358i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0209 + 0.118i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-5.10 - 1.85i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.50 + 2.94i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.38 - 4.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.294 - 0.509i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.35 - 7.67i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.88 + 3.25i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.52 - 8.62i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.79 - 4.85i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.22 - 0.446i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.419 - 2.37i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 + (-0.0412 + 0.0150i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.77 + 10.0i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.42 + 1.19i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.25 + 5.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.11 + 10.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.538 + 0.451i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.19 + 4.35i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (3.42 + 5.93i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.48 - 3.08i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.77911189290386337837027357524, −9.694877016626056811129066244388, −8.907203011846291828069083423007, −7.83880713209778634148588534043, −7.31885012368541213328025742531, −6.18970038323726849575907164364, −4.88380096434898959123177154531, −3.99041660046201896042605058887, −3.25652511580256522864492522189, −1.82673490893327176674171682593,
0.68922420167391306017141400706, 2.34869775546885080896416483899, 4.14975931807002050114568750055, 4.50708655410238575967695781925, 5.80488238314129573156390821527, 6.62663603172062296423306057588, 7.28345858536868011649960016816, 8.673755730065699479430204036894, 9.298401467369205293567239926664, 10.07738351216478191808013080409
