| L(s) = 1 | + (−1.20 + 0.439i)2-s + (−0.266 + 0.223i)4-s + (0.0775 + 0.439i)5-s + (−2.70 − 2.27i)7-s + (1.50 − 2.61i)8-s + (−0.286 − 0.497i)10-s + (−0.482 + 2.73i)11-s + (−3.09 − 1.12i)13-s + (4.26 + 1.55i)14-s + (−0.553 + 3.13i)16-s + (3.51 + 6.09i)17-s + (2.59 − 4.49i)19-s + (−0.118 − 0.0996i)20-s + (−0.620 − 3.51i)22-s + (5.57 − 4.67i)23-s + ⋯ |
| L(s) = 1 | + (−0.854 + 0.310i)2-s + (−0.133 + 0.111i)4-s + (0.0346 + 0.196i)5-s + (−1.02 − 0.858i)7-s + (0.533 − 0.923i)8-s + (−0.0907 − 0.157i)10-s + (−0.145 + 0.825i)11-s + (−0.857 − 0.312i)13-s + (1.14 + 0.415i)14-s + (−0.138 + 0.784i)16-s + (0.853 + 1.47i)17-s + (0.594 − 1.03i)19-s + (−0.0265 − 0.0222i)20-s + (−0.132 − 0.750i)22-s + (1.16 − 0.975i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679275 + 0.225087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679275 + 0.225087i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.20 - 0.439i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.0775 - 0.439i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.70 + 2.27i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.482 - 2.73i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (3.09 + 1.12i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.51 - 6.09i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 + 4.49i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.57 + 4.67i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (3.40 - 1.23i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.48 - 1.24i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.61 - 2.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.56 - 1.66i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1 - 5.67i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.31 - 1.93i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 8.77T + 53T^{2} \) |
| 59 | \( 1 + (-0.514 - 2.91i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.04 - 5.06i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.86 - 3.22i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.65 + 4.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.777 + 1.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.1 + 4.07i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-15.2 + 5.56i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (9.21 - 15.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.75 + 9.96i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.23613214312298838295366406632, −9.661747669377858401354812157025, −8.857157595068361199209231254682, −7.78832107290762916404529719991, −7.12708747277902394542599839353, −6.53040784018095790970305354993, −4.99847828683696960396067708761, −3.96293570155094689995839303667, −2.86533934677251544714233457288, −0.856851872084023224857245447313,
0.75673667439372483522270822151, 2.42153069496494517170434371515, 3.46117418656772904146565702428, 5.26561989792676168036002211020, 5.54207975453760841746800612663, 7.03681072558805266378757326664, 7.86577905320871287402887953761, 9.018239789769627059613325031256, 9.378252914789455179128444531897, 9.975797277097905897177038779843
