| L(s) = 1 | + (0.730 − 0.266i)2-s + (−1.06 + 0.896i)4-s + (−0.412 − 2.34i)5-s + (−1.91 − 1.60i)7-s + (−1.32 + 2.28i)8-s + (−0.924 − 1.60i)10-s + (−0.545 + 3.09i)11-s + (−1.25 − 0.457i)13-s + (−1.82 − 0.665i)14-s + (0.127 − 0.725i)16-s + (3.13 + 5.43i)17-s + (−4.03 + 6.98i)19-s + (2.53 + 2.13i)20-s + (0.424 + 2.40i)22-s + (−3.10 + 2.60i)23-s + ⋯ |
| L(s) = 1 | + (0.516 − 0.188i)2-s + (−0.534 + 0.448i)4-s + (−0.184 − 1.04i)5-s + (−0.724 − 0.607i)7-s + (−0.466 + 0.808i)8-s + (−0.292 − 0.506i)10-s + (−0.164 + 0.932i)11-s + (−0.348 − 0.126i)13-s + (−0.488 − 0.177i)14-s + (0.0319 − 0.181i)16-s + (0.760 + 1.31i)17-s + (−0.925 + 1.60i)19-s + (0.567 + 0.476i)20-s + (0.0904 + 0.512i)22-s + (−0.647 + 0.543i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.216006 + 0.430103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.216006 + 0.430103i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.730 + 0.266i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.412 + 2.34i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.91 + 1.60i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.545 - 3.09i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.25 + 0.457i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.13 - 5.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.03 - 6.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.10 - 2.60i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (8.72 - 3.17i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.16 + 1.82i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.76 + 4.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.67 + 2.43i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.405 + 2.30i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.53 - 2.96i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 0.135T + 53T^{2} \) |
| 59 | \( 1 + (-0.694 - 3.93i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.261 - 0.219i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (9.51 + 3.46i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.09 - 7.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.83 - 1.39i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.858 + 0.312i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (1.86 - 3.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.04 - 5.90i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.52999338515576202984355530164, −9.876491273837164823636877251256, −8.955211628925385073382777976378, −8.106719022479770449322854824862, −7.43475121912657064111121211044, −5.99544616257096618442643450494, −5.16372928594026477071824056352, −4.03456727242128250453172761060, −3.66703944305784590563584879475, −1.84116869113671053804583882412,
0.20384229118172381355196486345, 2.66932002829509761602312119378, 3.40550583553992846668692452740, 4.69516601661225939979425052622, 5.65769472827776126163382265337, 6.47427185993012610800712513389, 7.14740073862931848190060121442, 8.491276627361428042047997152842, 9.345055344016255946675315491366, 10.02174934031569449172020849922
