| 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
−9.924065762581204404109995840961, −9.270974772665873069150589364265, −8.773337132367160925953264622276, −6.92286434521599150807554597187, −6.45009375567097144237816215979, −5.68465427477886190665704188811, −4.69965270519434026230109352484, −3.56539122764417385581346551746, −2.58627373049647282635859256052, −0.38789444558534702368145110427,
2.29343099874461270630545993471, 3.08836825836788948344811182158, 4.25981010266686578879303816744, 5.37270659794521043294375590775, 5.95354283607225152160543342788, 7.06236143613110793203454573887, 8.071882215666778286191319150736, 9.325852166192781759211500777985, 9.919950853038765892124076354618, 10.27411684413931636283232969877
