L(s) = 1 | + (1 − 1.73i)2-s + (−0.999 − 1.73i)4-s + (1 − 1.73i)5-s + (−1.99 − 3.46i)10-s + (−1 − 1.73i)11-s − 13-s + (1.99 − 3.46i)16-s + (0.5 − 0.866i)19-s − 3.99·20-s − 3.99·22-s + (0.500 + 0.866i)25-s + (−1 + 1.73i)26-s − 4·29-s + (4.5 + 7.79i)31-s + (−3.99 − 6.92i)32-s + ⋯ |
L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.447 − 0.774i)5-s + (−0.632 − 1.09i)10-s + (−0.301 − 0.522i)11-s − 0.277·13-s + (0.499 − 0.866i)16-s + (0.114 − 0.198i)19-s − 0.894·20-s − 0.852·22-s + (0.100 + 0.173i)25-s + (−0.196 + 0.339i)26-s − 0.742·29-s + (0.808 + 1.39i)31-s + (−0.707 − 1.22i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.947827 - 1.91199i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.947827 - 1.91199i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.88381478687683212611247892420, −10.19108765328872676940965825392, −9.233153361206338198543210490442, −8.300831912792073410911994983002, −7.01842397817773169064687649106, −5.52760332090179227702685584001, −4.90003992433324134215765779852, −3.70354457972616565365145955591, −2.56147766452243104873584991151, −1.23016460075341789433949643971,
2.27689263339679432504867999504, 3.81829038348525041566569393420, 4.96952953562237227176871892073, 5.87295572801602683615539219319, 6.72508313428073797690881419585, 7.43218123963546323519361995834, 8.345616257938071359097484650533, 9.722738693879243613827129138377, 10.42037856594733381649778852369, 11.47427252918429605360581773829