| 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
| \(L(1)\) |
\(\approx\) |
\(0.324980 + 0.365177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324980 + 0.365177i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.33204495844135334674222053885, −9.874995530733242521257944439782, −9.274790020067099075173210922834, −8.026723673794208206349657460727, −7.30126719875965527464790614416, −5.94728066321944684804184588401, −5.41280865383068096317350459317, −4.30939806737114981744371994884, −2.60181095069118374609292220189, −1.75889105885540921563889366668,
0.26608268006152217468197900714, 2.54861434003666281299829179126, 3.35396172975016057350882524017, 4.94156898358211591995781494905, 5.55013499966117989175029724651, 6.96647146256944047007857556073, 7.60619713977163778282602877415, 8.380778147559506919972383086834, 9.183228273784771291061720692030, 10.13216686885463746326895510481
