| L(s) = 1 | + (−0.984 + 0.826i)2-s + (−0.0603 + 0.342i)4-s + (0.419 − 0.152i)5-s + (−0.613 − 3.47i)7-s + (−1.50 − 2.61i)8-s + (−0.286 + 0.497i)10-s + (−2.61 − 0.950i)11-s + (2.52 + 2.11i)13-s + (3.47 + 2.91i)14-s + (2.99 + 1.08i)16-s + (−3.51 + 6.09i)17-s + (2.59 + 4.49i)19-s + (0.0269 + 0.152i)20-s + (3.35 − 1.22i)22-s + (−1.26 + 7.16i)23-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.584i)2-s + (−0.0301 + 0.171i)4-s + (0.187 − 0.0682i)5-s + (−0.231 − 1.31i)7-s + (−0.533 − 0.923i)8-s + (−0.0907 + 0.157i)10-s + (−0.787 − 0.286i)11-s + (0.699 + 0.586i)13-s + (0.929 + 0.780i)14-s + (0.748 + 0.272i)16-s + (−0.853 + 1.47i)17-s + (0.594 + 1.03i)19-s + (0.00602 + 0.0341i)20-s + (0.716 − 0.260i)22-s + (−0.263 + 1.49i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.187349 + 0.565388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187349 + 0.565388i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.984 - 0.826i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.419 + 0.152i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.613 + 3.47i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.61 + 0.950i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.52 - 2.11i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.51 - 6.09i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 - 4.49i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.26 - 7.16i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.77 - 2.32i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.336 - 1.90i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.72 - 3.12i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.41 - 1.96i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.524 + 2.97i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 + (-2.78 + 1.01i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.36 - 7.76i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.23 + 6.06i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.65 + 4.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.777 - 1.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.11 - 7.65i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-12.4 + 10.4i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-9.21 - 15.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.50 + 3.45i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.57088056744978871173162603368, −9.732324733703669657768551143507, −8.986146463602879257885360622671, −7.917355443671308891493210481791, −7.55033160971440011817195290225, −6.49981603140403764605764604520, −5.70599467153838115025870090292, −4.04309551944051658215091583806, −3.51709311599819009683333200163, −1.46774027654399448641297869259,
0.39261155538447602912505401178, 2.33426981159199098935357561013, 2.74315261172861644744302697827, 4.72404151950754550304285022053, 5.59070276655723808050203010559, 6.36903299522922166515682988822, 7.72623881501398689010869637688, 8.678708613848956609573626491945, 9.260049826913159849001501968067, 9.943917588752662961698218192246
