| L(s) = 1 | + (2.23 − 1.46i)2-s + (2.02 − 4.70i)4-s + (−1.24 + 1.32i)5-s + (2.56 + 0.299i)7-s + (−1.44 − 8.21i)8-s + (−0.841 + 4.77i)10-s + (1.04 − 3.50i)11-s + (−0.0220 + 0.377i)13-s + (6.15 − 3.09i)14-s + (−8.24 − 8.73i)16-s + (2.13 − 0.778i)17-s + (−1.32 − 0.480i)19-s + (3.68 + 8.54i)20-s + (−2.80 − 9.35i)22-s + (−3.51 + 0.410i)23-s + ⋯ |
| L(s) = 1 | + (1.57 − 1.03i)2-s + (1.01 − 2.35i)4-s + (−0.557 + 0.590i)5-s + (0.969 + 0.113i)7-s + (−0.512 − 2.90i)8-s + (−0.266 + 1.50i)10-s + (0.316 − 1.05i)11-s + (−0.00610 + 0.104i)13-s + (1.64 − 0.826i)14-s + (−2.06 − 2.18i)16-s + (0.518 − 0.188i)17-s + (−0.303 − 0.110i)19-s + (0.824 + 1.91i)20-s + (−0.597 − 1.99i)22-s + (−0.733 + 0.0856i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.25543 - 2.83537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25543 - 2.83537i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.23 + 1.46i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (1.24 - 1.32i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-2.56 - 0.299i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-1.04 + 3.50i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (0.0220 - 0.377i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-2.13 + 0.778i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (1.32 + 0.480i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (3.51 - 0.410i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (5.90 + 2.96i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-5.83 - 7.83i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (0.463 + 0.389i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-2.79 - 1.84i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (-2.59 + 0.615i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-0.0900 + 0.120i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (6.72 - 11.6i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.21 - 4.06i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-2.95 - 6.84i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (-9.01 + 4.52i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (0.305 - 1.73i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.930 - 5.27i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (3.55 - 2.33i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (2.17 - 1.43i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (0.553 + 3.14i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (10.5 + 11.1i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−10.73430244033000688021602786384, −9.694458034240851226134740888731, −8.419656573196240459350797713677, −7.30581374666907893814703841879, −6.19595277160720561555764320711, −5.43456376760365461187482845999, −4.41726785986446557948930759248, −3.59368537655294846030217673950, −2.69157398262830543739530743220, −1.35354942172349466088366936233,
2.10574331548997128217869221050, 3.73221801673635632998837666688, 4.42578084109160312244959025870, 5.06615191029430478796631612773, 6.05499799150968308389517429966, 7.04856671591324988440442489476, 7.947110012789529718473740796304, 8.242906853990808936034112968332, 9.680685889351894840255127898795, 11.11256167423317394060714801876
