| L(s) = 1 | + (−0.274 + 1.55i)2-s + (−0.461 − 0.168i)4-s + (1.28 − 1.07i)5-s + (−2.61 + 0.950i)7-s + (−1.19 + 2.06i)8-s + (1.32 + 2.29i)10-s + (−3.18 − 2.66i)11-s + (1.19 + 6.77i)13-s + (−0.761 − 4.32i)14-s + (−3.63 − 3.04i)16-s + (−0.488 − 0.845i)17-s + (−1.34 + 2.32i)19-s + (−0.775 + 0.282i)20-s + (5.01 − 4.21i)22-s + (−1.51 − 0.551i)23-s + ⋯ |
| L(s) = 1 | + (−0.193 + 1.09i)2-s + (−0.230 − 0.0840i)4-s + (0.575 − 0.482i)5-s + (−0.987 + 0.359i)7-s + (−0.420 + 0.729i)8-s + (0.418 + 0.725i)10-s + (−0.958 − 0.804i)11-s + (0.331 + 1.87i)13-s + (−0.203 − 1.15i)14-s + (−0.908 − 0.761i)16-s + (−0.118 − 0.205i)17-s + (−0.308 + 0.533i)19-s + (−0.173 + 0.0630i)20-s + (1.07 − 0.897i)22-s + (−0.316 − 0.115i)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
| \(L(1)\) |
\(\approx\) |
\(0.0518245 - 0.889792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0518245 - 0.889792i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.274 - 1.55i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.28 + 1.07i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.61 - 0.950i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (3.18 + 2.66i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.19 - 6.77i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.488 + 0.845i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.34 - 2.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.51 + 0.551i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.42 - 8.10i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.981 + 0.357i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.654 - 1.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.841 + 4.77i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.53 - 6.32i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (11.7 - 4.27i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + (-6.93 + 5.81i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.20 + 0.439i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.806 + 4.57i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.81 + 4.87i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.28 + 3.95i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.808 + 4.58i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.00 + 5.67i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (2.27 - 3.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.56 - 5.51i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.80357803240610024841242100911, −9.537283630377282556244603192276, −9.041638826187641327373809236566, −8.299093986117809394878546512485, −7.21426327700109366007969702245, −6.37333285651062967531135126534, −5.82427446274146776187585369694, −4.86484726558361030930049400877, −3.33032527192943123321799127403, −2.02162826268360403663581031228,
0.44633368415480762564531086022, 2.27267344316694506914769251376, 2.92708479418894967932207629037, 3.98935093615007658013191309266, 5.54121859705020867348576479413, 6.37168927317150650663303020699, 7.30286153185360907151782400350, 8.361875661153851825495160548541, 9.682304816420932476876267034011, 10.15783097901399190417885463381
