| L(s) = 1 | + (−0.369 − 2.09i)2-s + (−2.36 + 0.860i)4-s + (−1.58 − 1.33i)5-s + (−4.55 − 1.65i)7-s + (0.547 + 0.949i)8-s + (−2.20 + 3.81i)10-s + (3.17 − 2.66i)11-s + (−0.211 + 1.19i)13-s + (−1.78 + 10.1i)14-s + (−2.07 + 1.73i)16-s + (−1.18 + 2.04i)17-s + (0.919 + 1.59i)19-s + (4.89 + 1.78i)20-s + (−6.75 − 5.66i)22-s + (4.04 − 1.47i)23-s + ⋯ |
| L(s) = 1 | + (−0.260 − 1.47i)2-s + (−1.18 + 0.430i)4-s + (−0.709 − 0.595i)5-s + (−1.72 − 0.626i)7-s + (0.193 + 0.335i)8-s + (−0.695 + 1.20i)10-s + (0.958 − 0.804i)11-s + (−0.0585 + 0.332i)13-s + (−0.478 + 2.71i)14-s + (−0.517 + 0.434i)16-s + (−0.286 + 0.496i)17-s + (0.210 + 0.365i)19-s + (1.09 + 0.398i)20-s + (−1.44 − 1.20i)22-s + (0.843 − 0.306i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.150142 + 0.0754043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.150142 + 0.0754043i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.369 + 2.09i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (1.58 + 1.33i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (4.55 + 1.65i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-3.17 + 2.66i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.211 - 1.19i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.18 - 2.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.919 - 1.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.04 + 1.47i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.517 - 2.93i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.38 - 0.503i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (4.48 - 7.76i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.392 + 2.22i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.20 - 3.52i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (6.74 + 2.45i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + (-0.200 - 0.168i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.18 - 1.52i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.717 + 4.06i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (1.54 - 2.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.38 + 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.790 - 4.48i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.46 + 8.32i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (8.48 + 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.91 - 3.28i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.812272763921003674862695321583, −9.048554636387199665733850950759, −8.420251976356060367291027817949, −6.92195211753311840477066550446, −6.27012164963051385941841195433, −4.53450180413828336705330397244, −3.61810751726583474565997914313, −3.13476788024626796981938175237, −1.31411218524151567086627722720, −0.10152027922406283723407045502,
2.76535115410649796126540108189, 3.81617218767573750309222849637, 5.17748956215765357775370880409, 6.16357944395494992509786132726, 6.98626963704575324823222721684, 7.19086447919229797279704163493, 8.454861784040531506707389095565, 9.378195951294963971246387543876, 9.693853229268377468209145583638, 11.12649494962841536033886159098
