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 | \( 1 + (-5.63 + 2.05i)T + (74.3 - 62.3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.44541500893455452968470109896, −9.678591651941488454060116965278, −8.771850998591069058229587896995, −8.295390843003481429119630043123, −6.40350005639763292533983716716, −6.11401366603186829898316977723, −5.30953432556001736601671239957, −3.73225217371856068762707113506, −2.33527865993842520642012374121, −1.44729158757537163438637765762,
0.936400010324329930953082964804, 2.65814116663456856482631036333, 3.96344916093681313247145679996, 4.84691888951809120299817062953, 6.27414932732651310538599437534, 6.90115979567280064570710520838, 7.73430791026173315250009501125, 8.787693487677879135311652039168, 9.533424763107087925591567190915, 9.919673379725125716905693482918