| L(s) = 1 | + (1.83 + 0.510i)2-s + (1.39 + 0.841i)4-s + (−2.94 + 0.114i)5-s + (−2.67 − 0.417i)7-s + (−0.484 − 0.513i)8-s + (−5.46 − 1.29i)10-s + (−0.589 − 4.31i)11-s + (1.84 − 2.69i)13-s + (−4.68 − 2.13i)14-s + (−2.14 − 4.07i)16-s + (−4.70 + 2.36i)17-s + (0.0335 + 0.575i)19-s + (−4.20 − 2.32i)20-s + (1.12 − 8.21i)22-s + (−2.84 + 7.36i)23-s + ⋯ |
| L(s) = 1 | + (1.29 + 0.361i)2-s + (0.697 + 0.420i)4-s + (−1.31 + 0.0511i)5-s + (−1.00 − 0.157i)7-s + (−0.171 − 0.181i)8-s + (−1.72 − 0.409i)10-s + (−0.177 − 1.30i)11-s + (0.511 − 0.746i)13-s + (−1.25 − 0.569i)14-s + (−0.536 − 1.01i)16-s + (−1.14 + 0.572i)17-s + (0.00769 + 0.132i)19-s + (−0.940 − 0.519i)20-s + (0.239 − 1.75i)22-s + (−0.592 + 1.53i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.399009 - 0.728388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.399009 - 0.728388i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.83 - 0.510i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (2.94 - 0.114i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (2.67 + 0.417i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.589 + 4.31i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-1.84 + 2.69i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (4.70 - 2.36i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.0335 - 0.575i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (2.84 - 7.36i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-0.495 - 0.353i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-5.02 + 5.75i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-1.86 - 2.50i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-2.17 + 8.45i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-0.623 - 0.565i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (5.83 + 6.68i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (2.35 - 13.3i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (6.01 - 2.45i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-4.92 + 2.97i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (7.89 - 5.64i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (0.635 + 2.12i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (5.07 - 1.20i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-8.20 + 8.04i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (3.22 + 12.5i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (1.18 - 3.94i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (4.78 + 0.185i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.29976204939121383367561171248, −9.111136949166655233518736873257, −8.158749230173064021925089954067, −7.32751850478936638643202538130, −6.22798994224766678585922646448, −5.77065207579773059229778344192, −4.38404617777074681005215367667, −3.63420687764808119394126732497, −3.07320540737956303202203609442, −0.26576634062613349735267451467,
2.35919205136858386488206651789, 3.36014696114512785849730245534, 4.43738371910145522110474319995, 4.66751561536893385813620328753, 6.36069177766074974930789357329, 6.80370921282931591436321959975, 8.069847142707966981507871510442, 8.959119447884363859958893001523, 9.959441970844358182300496917015, 11.12416451383753354706521747290
