| L(s) = 1 | + (−0.595 + 0.499i)2-s + (−0.242 + 1.37i)4-s + (2.23 − 0.812i)5-s + (−0.434 − 2.46i)7-s + (−1.32 − 2.28i)8-s + (−0.924 + 1.60i)10-s + (2.95 + 1.07i)11-s + (1.02 + 0.859i)13-s + (1.49 + 1.25i)14-s + (−0.692 − 0.251i)16-s + (3.13 − 5.43i)17-s + (−4.03 − 6.98i)19-s + (0.575 + 3.26i)20-s + (−2.29 + 0.835i)22-s + (−0.704 + 3.99i)23-s + ⋯ |
| L(s) = 1 | + (−0.421 + 0.353i)2-s + (−0.121 + 0.686i)4-s + (0.998 − 0.363i)5-s + (−0.164 − 0.931i)7-s + (−0.466 − 0.808i)8-s + (−0.292 + 0.506i)10-s + (0.890 + 0.323i)11-s + (0.283 + 0.238i)13-s + (0.398 + 0.334i)14-s + (−0.173 − 0.0629i)16-s + (0.760 − 1.31i)17-s + (−0.925 − 1.60i)19-s + (0.128 + 0.730i)20-s + (−0.489 + 0.178i)22-s + (−0.146 + 0.832i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38349 - 0.0805795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38349 - 0.0805795i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.595 - 0.499i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-2.23 + 0.812i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.434 + 2.46i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.95 - 1.07i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.02 - 0.859i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.03 + 6.98i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.704 - 3.99i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.11 + 5.96i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.491 + 2.78i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.76 - 4.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.44 - 4.56i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.19 + 0.799i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.801 - 4.54i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 0.135T + 53T^{2} \) |
| 59 | \( 1 + (3.75 - 1.36i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.0593 - 0.336i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.75 - 6.50i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.09 + 7.09i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 - 10.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.12 + 2.62i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.699 - 0.587i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (1.86 + 3.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.63 - 2.05i)T + (74.3 + 62.3i)T^{2} \) |
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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
−9.919673379725125716905693482918, −9.533424763107087925591567190915, −8.787693487677879135311652039168, −7.73430791026173315250009501125, −6.90115979567280064570710520838, −6.27414932732651310538599437534, −4.84691888951809120299817062953, −3.96344916093681313247145679996, −2.65814116663456856482631036333, −0.936400010324329930953082964804,
1.44729158757537163438637765762, 2.33527865993842520642012374121, 3.73225217371856068762707113506, 5.30953432556001736601671239957, 6.11401366603186829898316977723, 6.40350005639763292533983716716, 8.295390843003481429119630043123, 8.771850998591069058229587896995, 9.678591651941488454060116965278, 10.44541500893455452968470109896
