L(s) = 1 | + (−0.826 − 0.300i)2-s + (−0.939 − 0.788i)4-s + (−0.673 + 3.82i)5-s + (−1.67 + 1.40i)7-s + (1.41 + 2.45i)8-s + (1.70 − 2.95i)10-s + (−0.0282 − 0.160i)11-s + (−2.26 + 0.824i)13-s + (1.80 − 0.657i)14-s + (−0.00727 − 0.0412i)16-s + (1.5 − 2.59i)17-s + (−1.79 − 3.11i)19-s + (3.64 − 3.05i)20-s + (−0.0248 + 0.140i)22-s + (−2.17 − 1.82i)23-s + ⋯ |
L(s) = 1 | + (−0.584 − 0.212i)2-s + (−0.469 − 0.394i)4-s + (−0.301 + 1.70i)5-s + (−0.632 + 0.530i)7-s + (0.501 + 0.868i)8-s + (0.539 − 0.934i)10-s + (−0.00850 − 0.0482i)11-s + (−0.628 + 0.228i)13-s + (0.482 − 0.175i)14-s + (−0.00181 − 0.0103i)16-s + (0.363 − 0.630i)17-s + (−0.412 − 0.714i)19-s + (0.815 − 0.683i)20-s + (−0.00529 + 0.0300i)22-s + (−0.453 − 0.380i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.826 + 0.300i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.673 - 3.82i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.67 - 1.40i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0282 + 0.160i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.26 - 0.824i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.79 + 3.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.17 + 1.82i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.31 - 2.29i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.97 + 3.33i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.31 + 5.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.45 - 1.98i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.08 + 6.13i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.66 - 4.75i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + (-0.889 + 5.04i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.89 - 2.43i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.51 + 2.00i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (7.65 - 13.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.34 + 7.51i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 - 0.433i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (7.96 + 2.89i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.86 - 6.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.678 + 3.84i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.988363607939132069746781167456, −9.449415743606584889137959980691, −8.430189688858237448394770492402, −7.40677856036232534594574792995, −6.64537503309464185065040199034, −5.72036925696362439667627769453, −4.47179218328817879593997834614, −3.11808074763030258520557945789, −2.27231113389203457391997785840, 0,
1.32301700358388038993176109676, 3.50678411776986203598025252829, 4.35801853833068107226980476221, 5.18012185744780377033439776125, 6.47497593427075594051643364878, 7.67841427085957292690811578498, 8.219735563431044179513054499588, 8.898123839623596293365820264817, 9.831578632690819717348629785288