L(s) = 1 | + (1.04 − 0.290i)2-s + (−0.704 + 0.425i)4-s + (2.06 + 0.0803i)5-s + (−4.16 + 0.651i)7-s + (−2.10 + 2.22i)8-s + (2.18 − 0.518i)10-s + (−0.767 + 5.61i)11-s + (−3.39 − 4.94i)13-s + (−4.16 + 1.89i)14-s + (−0.781 + 1.48i)16-s + (−1.18 − 0.593i)17-s + (−0.383 + 6.58i)19-s + (−1.49 + 0.823i)20-s + (0.832 + 6.09i)22-s + (0.756 + 1.96i)23-s + ⋯ |
L(s) = 1 | + (0.739 − 0.205i)2-s + (−0.352 + 0.212i)4-s + (0.925 + 0.0359i)5-s + (−1.57 + 0.246i)7-s + (−0.743 + 0.787i)8-s + (0.691 − 0.163i)10-s + (−0.231 + 1.69i)11-s + (−0.940 − 1.37i)13-s + (−1.11 + 0.505i)14-s + (−0.195 + 0.371i)16-s + (−0.286 − 0.144i)17-s + (−0.0879 + 1.50i)19-s + (−0.333 + 0.184i)20-s + (0.177 + 1.29i)22-s + (0.157 + 0.408i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.477484 + 0.906010i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.477484 + 0.906010i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.04 + 0.290i)T + (1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (-2.06 - 0.0803i)T + (4.98 + 0.387i)T^{2} \) |
| 7 | \( 1 + (4.16 - 0.651i)T + (6.66 - 2.13i)T^{2} \) |
| 11 | \( 1 + (0.767 - 5.61i)T + (-10.5 - 2.94i)T^{2} \) |
| 13 | \( 1 + (3.39 + 4.94i)T + (-4.68 + 12.1i)T^{2} \) |
| 17 | \( 1 + (1.18 + 0.593i)T + (10.1 + 13.6i)T^{2} \) |
| 19 | \( 1 + (0.383 - 6.58i)T + (-18.8 - 2.20i)T^{2} \) |
| 23 | \( 1 + (-0.756 - 1.96i)T + (-17.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (-3.30 + 2.36i)T + (9.38 - 27.4i)T^{2} \) |
| 31 | \( 1 + (-0.0367 - 0.0420i)T + (-4.19 + 30.7i)T^{2} \) |
| 37 | \( 1 + (2.13 - 2.87i)T + (-10.6 - 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.587 - 2.28i)T + (-35.8 + 19.8i)T^{2} \) |
| 43 | \( 1 + (-2.40 + 2.18i)T + (4.16 - 42.7i)T^{2} \) |
| 47 | \( 1 + (0.143 - 0.164i)T + (-6.36 - 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.06 - 6.01i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.683 + 0.279i)T + (42.1 + 41.3i)T^{2} \) |
| 61 | \( 1 + (-11.3 - 6.83i)T + (28.4 + 53.9i)T^{2} \) |
| 67 | \( 1 + (4.99 + 3.56i)T + (21.6 + 63.3i)T^{2} \) |
| 71 | \( 1 + (3.11 - 10.4i)T + (-59.3 - 39.0i)T^{2} \) |
| 73 | \( 1 + (1.60 + 0.380i)T + (65.2 + 32.7i)T^{2} \) |
| 79 | \( 1 + (0.413 + 0.405i)T + (1.53 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-0.994 + 3.85i)T + (-72.6 - 40.0i)T^{2} \) |
| 89 | \( 1 + (-2.78 - 9.29i)T + (-74.3 + 48.9i)T^{2} \) |
| 97 | \( 1 + (6.85 - 0.266i)T + (96.7 - 7.51i)T^{2} \) |
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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
−10.27411684413931636283232969877, −9.919950853038765892124076354618, −9.325852166192781759211500777985, −8.071882215666778286191319150736, −7.06236143613110793203454573887, −5.95354283607225152160543342788, −5.37270659794521043294375590775, −4.25981010266686578879303816744, −3.08836825836788948344811182158, −2.29343099874461270630545993471,
0.38789444558534702368145110427, 2.58627373049647282635859256052, 3.56539122764417385581346551746, 4.69965270519434026230109352484, 5.68465427477886190665704188811, 6.45009375567097144237816215979, 6.92286434521599150807554597187, 8.773337132367160925953264622276, 9.270974772665873069150589364265, 9.924065762581204404109995840961