| 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 | \( 1 + (0.245 + 0.0892i)T + (74.3 + 62.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11153076714323134271805246612, −9.462725106196484380060760694494, −9.081606645956273057843665750489, −8.032561134728158956589448015318, −7.28301414536710830338069755847, −6.26468813086857699810969965636, −5.55472309760831013214270384438, −4.24958985855286868250437333755, −2.91645024259358726082734741659, −1.39664286449151233890591032924,
0.930330046729912974723155318378, 1.97915308860002905321069485065, 3.38198641614930659633633537006, 4.61309373298222874294605468836, 5.83930849650861587454565100009, 6.66406482000830156000464477687, 7.63925676254155276605160078338, 8.908822741580011014177347684266, 9.407255981142734151573037191456, 10.06034536672924034009620512257
