| L(s) = 1 | + (−0.126 + 2.17i)2-s + (−2.71 − 0.317i)4-s + (0.629 − 0.149i)5-s + (−1.09 − 1.46i)7-s + (0.277 − 1.57i)8-s + (0.244 + 1.38i)10-s + (0.830 − 0.880i)11-s + (−4.76 − 2.39i)13-s + (3.32 − 2.18i)14-s + (−1.93 − 0.459i)16-s + (−6.43 − 2.34i)17-s + (−5.97 + 2.17i)19-s + (−1.75 + 0.205i)20-s + (1.80 + 1.91i)22-s + (−1.86 + 2.49i)23-s + ⋯ |
| L(s) = 1 | + (−0.0894 + 1.53i)2-s + (−1.35 − 0.158i)4-s + (0.281 − 0.0666i)5-s + (−0.412 − 0.554i)7-s + (0.0980 − 0.556i)8-s + (0.0772 + 0.438i)10-s + (0.250 − 0.265i)11-s + (−1.32 − 0.663i)13-s + (0.888 − 0.584i)14-s + (−0.484 − 0.114i)16-s + (−1.56 − 0.567i)17-s + (−1.37 + 0.499i)19-s + (−0.392 + 0.0458i)20-s + (0.385 + 0.408i)22-s + (−0.388 + 0.521i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0105539 - 0.00895022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0105539 - 0.00895022i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.126 - 2.17i)T + (-1.98 - 0.232i)T^{2} \) |
| 5 | \( 1 + (-0.629 + 0.149i)T + (4.46 - 2.24i)T^{2} \) |
| 7 | \( 1 + (1.09 + 1.46i)T + (-2.00 + 6.70i)T^{2} \) |
| 11 | \( 1 + (-0.830 + 0.880i)T + (-0.639 - 10.9i)T^{2} \) |
| 13 | \( 1 + (4.76 + 2.39i)T + (7.76 + 10.4i)T^{2} \) |
| 17 | \( 1 + (6.43 + 2.34i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (5.97 - 2.17i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (1.86 - 2.49i)T + (-6.59 - 22.0i)T^{2} \) |
| 29 | \( 1 + (-4.91 - 3.23i)T + (11.4 + 26.6i)T^{2} \) |
| 31 | \( 1 + (1.09 + 2.54i)T + (-21.2 + 22.5i)T^{2} \) |
| 37 | \( 1 + (1.09 - 0.918i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (0.0648 + 1.11i)T + (-40.7 + 4.75i)T^{2} \) |
| 43 | \( 1 + (-2.76 - 9.22i)T + (-35.9 + 23.6i)T^{2} \) |
| 47 | \( 1 + (-2.41 + 5.60i)T + (-32.2 - 34.1i)T^{2} \) |
| 53 | \( 1 + (-4.26 - 7.38i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.43 + 1.51i)T + (-3.43 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-3.56 + 0.416i)T + (59.3 - 14.0i)T^{2} \) |
| 67 | \( 1 + (1.01 - 0.670i)T + (26.5 - 61.5i)T^{2} \) |
| 71 | \( 1 + (-1.41 - 8.02i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.11 + 6.32i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.829 + 14.2i)T + (-78.4 - 9.17i)T^{2} \) |
| 83 | \( 1 + (0.390 - 6.70i)T + (-82.4 - 9.63i)T^{2} \) |
| 89 | \( 1 + (2.70 - 15.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.62 + 0.860i)T + (86.6 + 43.5i)T^{2} \) |
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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
−9.917582259886488733597920123291, −9.142899954336744811486036252928, −8.289827539323591390556247770293, −7.42277803136993043317285383178, −6.73448807569398352898298227455, −5.97864813666891853631009992831, −4.99630688588970805268223574612, −4.08517173376634993390689783936, −2.40436014833916238713922052127, −0.00662445104802176071525774802,
2.16001641146265568213223219044, 2.41907928789678295983944234593, 4.04154064162285570732245970352, 4.66062332769725233734888176280, 6.22940408710995462021251030907, 6.94349977739220713058400258317, 8.536012861440570925257390196327, 9.125931088684521820551292484413, 9.923933872485054188598657198685, 10.56252509078250152114731104337
