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
−10.15783097901399190417885463381, −9.682304816420932476876267034011, −8.361875661153851825495160548541, −7.30286153185360907151782400350, −6.37168927317150650663303020699, −5.54121859705020867348576479413, −3.98935093615007658013191309266, −2.92708479418894967932207629037, −2.27267344316694506914769251376, −0.44633368415480762564531086022,
2.02162826268360403663581031228, 3.33032527192943123321799127403, 4.86484726558361030930049400877, 5.82427446274146776187585369694, 6.37333285651062967531135126534, 7.21426327700109366007969702245, 8.299093986117809394878546512485, 9.041638826187641327373809236566, 9.537283630377282556244603192276, 10.80357803240610024841242100911