| 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
−11.12649494962841536033886159098, −9.693853229268377468209145583638, −9.378195951294963971246387543876, −8.454861784040531506707389095565, −7.19086447919229797279704163493, −6.98626963704575324823222721684, −6.16357944395494992509786132726, −5.17748956215765357775370880409, −3.81617218767573750309222849637, −2.76535115410649796126540108189,
0.10152027922406283723407045502, 1.31411218524151567086627722720, 3.13476788024626796981938175237, 3.61810751726583474565997914313, 4.53450180413828336705330397244, 6.27012164963051385941841195433, 6.92195211753311840477066550446, 8.420251976356060367291027817949, 9.048554636387199665733850950759, 9.812272763921003674862695321583
