| L(s) = 1 | + (0.629 − 0.691i)2-s + (−0.582 + 1.04i)3-s + (0.103 + 1.12i)4-s + (0.297 − 2.13i)5-s + (0.356 + 1.06i)6-s + (0.545 − 2.31i)7-s + (2.33 + 1.76i)8-s + (0.823 + 1.33i)9-s + (−1.28 − 1.55i)10-s + (2.23 + 1.85i)11-s + (−1.23 − 0.544i)12-s + (1.25 − 1.66i)13-s + (−1.25 − 1.83i)14-s + (2.06 + 1.55i)15-s + (0.473 − 0.0885i)16-s + (0.0158 + 4.12i)17-s + ⋯ |
| L(s) = 1 | + (0.445 − 0.488i)2-s + (−0.336 + 0.604i)3-s + (0.0519 + 0.560i)4-s + (0.133 − 0.955i)5-s + (0.145 + 0.433i)6-s + (0.206 − 0.875i)7-s + (0.824 + 0.622i)8-s + (0.274 + 0.443i)9-s + (−0.407 − 0.490i)10-s + (0.673 + 0.559i)11-s + (−0.356 − 0.157i)12-s + (0.349 − 0.462i)13-s + (−0.336 − 0.490i)14-s + (0.532 + 0.401i)15-s + (0.118 − 0.0221i)16-s + (0.00383 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64777 - 0.0115913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64777 - 0.0115913i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-0.0158 - 4.12i)T \) |
| good | 2 | \( 1 + (-0.629 + 0.691i)T + (-0.184 - 1.99i)T^{2} \) |
| 3 | \( 1 + (0.582 - 1.04i)T + (-1.57 - 2.55i)T^{2} \) |
| 5 | \( 1 + (-0.297 + 2.13i)T + (-4.80 - 1.36i)T^{2} \) |
| 7 | \( 1 + (-0.545 + 2.31i)T + (-6.26 - 3.12i)T^{2} \) |
| 11 | \( 1 + (-2.23 - 1.85i)T + (2.02 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 1.66i)T + (-3.55 - 12.5i)T^{2} \) |
| 19 | \( 1 + (-0.738 - 0.809i)T + (-1.75 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.40 + 5.97i)T + (-20.5 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.403 + 0.486i)T + (-5.32 - 28.5i)T^{2} \) |
| 31 | \( 1 + (5.02 - 0.700i)T + (29.8 - 8.48i)T^{2} \) |
| 37 | \( 1 + (0.571 + 1.29i)T + (-24.9 + 27.3i)T^{2} \) |
| 41 | \( 1 + (9.65 + 5.37i)T + (21.5 + 34.8i)T^{2} \) |
| 43 | \( 1 + (-1.24 + 6.67i)T + (-40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + (9.43 + 5.84i)T + (20.9 + 42.0i)T^{2} \) |
| 53 | \( 1 + (-1.72 - 2.78i)T + (-23.6 + 47.4i)T^{2} \) |
| 59 | \( 1 + (12.3 - 6.15i)T + (35.5 - 47.0i)T^{2} \) |
| 61 | \( 1 + (0.0346 + 0.103i)T + (-48.6 + 36.7i)T^{2} \) |
| 67 | \( 1 + (-0.731 + 0.666i)T + (6.18 - 66.7i)T^{2} \) |
| 71 | \( 1 + (-4.95 - 1.16i)T + (63.5 + 31.6i)T^{2} \) |
| 73 | \( 1 + (-3.12 - 2.13i)T + (26.3 + 68.0i)T^{2} \) |
| 79 | \( 1 + (0.186 - 4.03i)T + (-78.6 - 7.28i)T^{2} \) |
| 83 | \( 1 + (15.9 + 4.53i)T + (70.5 + 43.6i)T^{2} \) |
| 89 | \( 1 + (-5.88 - 7.79i)T + (-24.3 + 85.6i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 3.15i)T + (86.8 + 43.2i)T^{2} \) |
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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
−11.92358755387008248393409171566, −10.75236465659430332216338839613, −10.34899324408035684632811446768, −8.949844716254303801177644036385, −8.056758122881495265398232731591, −6.96826939661059862534274192621, −5.29281045584220253903160908973, −4.43839610565307799655343273755, −3.72597110239452383455797543145, −1.69451117203236731284946669931,
1.54685454227097880391478053956, 3.33574438452265425519069560846, 4.99117999693732879954540995161, 6.10205453168580568687557783904, 6.61879753377817357164764884411, 7.47907062011682020090928328791, 9.081637843909273857421524004375, 9.856851931051208509799592065378, 11.29097108729631767623281403824, 11.54998358591044071050480950067
