L(s) = 1 | + (−1.03 − 0.866i)2-s + (−0.0320 − 0.181i)4-s + (1.55 + 0.565i)5-s + (0.418 − 2.37i)7-s + (−1.47 + 2.54i)8-s + (−1.11 − 1.92i)10-s + (5.58 − 2.03i)11-s + (−2.47 + 2.07i)13-s + (−2.48 + 2.08i)14-s + (3.37 − 1.22i)16-s + (1.5 + 2.59i)17-s + (3.31 − 5.74i)19-s + (0.0530 − 0.300i)20-s + (−7.52 − 2.73i)22-s + (0.511 + 2.89i)23-s + ⋯ |
L(s) = 1 | + (−0.729 − 0.612i)2-s + (−0.0160 − 0.0909i)4-s + (0.694 + 0.252i)5-s + (0.158 − 0.897i)7-s + (−0.520 + 0.901i)8-s + (−0.352 − 0.609i)10-s + (1.68 − 0.612i)11-s + (−0.685 + 0.575i)13-s + (−0.665 + 0.558i)14-s + (0.844 − 0.307i)16-s + (0.363 + 0.630i)17-s + (0.761 − 1.31i)19-s + (0.0118 − 0.0672i)20-s + (−1.60 − 0.583i)22-s + (0.106 + 0.604i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.825755 - 0.875249i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.825755 - 0.875249i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.03 + 0.866i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-1.55 - 0.565i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.418 + 2.37i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-5.58 + 2.03i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.47 - 2.07i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 + 5.74i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.511 - 2.89i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.988 - 0.829i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.102 + 0.579i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (0.0209 + 0.0362i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.75 + 3.15i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.87 + 1.77i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.648 + 3.67i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + (6.90 + 2.51i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.91 + 10.8i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.42 + 1.19i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.75 + 4.77i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.77 - 4.81i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.89 + 2.43i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.05 + 2.56i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.07 + 7.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.245 - 0.0892i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.06034536672924034009620512257, −9.407255981142734151573037191456, −8.908822741580011014177347684266, −7.63925676254155276605160078338, −6.66406482000830156000464477687, −5.83930849650861587454565100009, −4.61309373298222874294605468836, −3.38198641614930659633633537006, −1.97915308860002905321069485065, −0.930330046729912974723155318378,
1.39664286449151233890591032924, 2.91645024259358726082734741659, 4.24958985855286868250437333755, 5.55472309760831013214270384438, 6.26468813086857699810969965636, 7.28301414536710830338069755847, 8.032561134728158956589448015318, 9.081606645956273057843665750489, 9.462725106196484380060760694494, 10.11153076714323134271805246612