L(s) = 1 | + (−1.88 − 1.57i)2-s + (0.701 + 3.97i)4-s + (2.89 + 1.05i)5-s + (−0.461 + 2.61i)7-s + (2.50 − 4.34i)8-s + (−3.78 − 6.55i)10-s + (3.22 − 1.17i)11-s + (−2.56 + 2.14i)13-s + (5.00 − 4.20i)14-s + (−3.98 + 1.45i)16-s + (−1.28 − 2.22i)17-s + (1.04 − 1.81i)19-s + (−2.16 + 12.2i)20-s + (−7.93 − 2.88i)22-s + (0.0928 + 0.526i)23-s + ⋯ |
L(s) = 1 | + (−1.33 − 1.11i)2-s + (0.350 + 1.98i)4-s + (1.29 + 0.471i)5-s + (−0.174 + 0.989i)7-s + (0.886 − 1.53i)8-s + (−1.19 − 2.07i)10-s + (0.973 − 0.354i)11-s + (−0.710 + 0.596i)13-s + (1.33 − 1.12i)14-s + (−0.997 + 0.363i)16-s + (−0.311 − 0.540i)17-s + (0.240 − 0.416i)19-s + (−0.483 + 2.74i)20-s + (−1.69 − 0.615i)22-s + (0.0193 + 0.109i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.918339 + 0.0534871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.918339 + 0.0534871i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.88 + 1.57i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-2.89 - 1.05i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.461 - 2.61i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.22 + 1.17i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.56 - 2.14i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.28 + 2.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.04 + 1.81i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0928 - 0.526i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.93 - 1.62i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.33 - 7.59i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.14 - 8.91i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.74 - 3.14i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.57 - 0.937i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.982 - 5.57i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 + (1.55 + 0.566i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.49 + 14.1i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.50 - 3.77i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (7.40 + 12.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.940 - 1.62i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.1 - 11.0i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.04 - 2.55i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.54 - 4.41i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.99 + 3.63i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.18001876403768337330256481192, −9.501166272542992747060230236365, −9.149081621774300019051950143535, −8.308424221859168420185774422349, −6.95104349968867602442955102627, −6.26129925953119011942009517017, −4.94075250479979663988661678620, −3.15516861636421851867962425119, −2.43792304623003983768801265251, −1.43393873286286976037114141155,
0.799544245176089800497542993092, 2.00881498712150820853862063334, 4.13904573191949809767614910064, 5.47930350056146681980484514084, 6.15624516192480244485645368271, 7.01683298780426057058586477563, 7.72860404893044803203137020450, 8.772306466511317301670800034894, 9.443798805723985563461869798385, 10.09166542760513932525248519539