| 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
−10.56252509078250152114731104337, −9.923933872485054188598657198685, −9.125931088684521820551292484413, −8.536012861440570925257390196327, −6.94349977739220713058400258317, −6.22940408710995462021251030907, −4.66062332769725233734888176280, −4.04154064162285570732245970352, −2.41907928789678295983944234593, −2.16001641146265568213223219044,
0.00662445104802176071525774802, 2.40436014833916238713922052127, 4.08517173376634993390689783936, 4.99630688588970805268223574612, 5.97864813666891853631009992831, 6.73448807569398352898298227455, 7.42277803136993043317285383178, 8.289827539323591390556247770293, 9.142899954336744811486036252928, 9.917582259886488733597920123291
