| L(s) = 1 | + (−0.155 + 0.428i)2-s + (0.859 − 2.87i)3-s + (2.90 + 2.43i)4-s + (−2.41 − 0.426i)5-s + (1.09 + 0.816i)6-s + (−3.31 + 2.78i)7-s + (−3.07 + 1.77i)8-s + (−7.52 − 4.94i)9-s + (0.559 − 0.969i)10-s + (−11.0 + 1.94i)11-s + (9.50 − 6.25i)12-s + (8.99 − 3.27i)13-s + (−0.674 − 1.85i)14-s + (−3.30 + 6.58i)15-s + (2.35 + 13.3i)16-s + (23.9 + 13.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.0779 + 0.214i)2-s + (0.286 − 0.958i)3-s + (0.726 + 0.609i)4-s + (−0.483 − 0.0852i)5-s + (0.182 + 0.136i)6-s + (−0.473 + 0.397i)7-s + (−0.384 + 0.222i)8-s + (−0.835 − 0.548i)9-s + (0.0559 − 0.0969i)10-s + (−1.00 + 0.176i)11-s + (0.791 − 0.521i)12-s + (0.692 − 0.251i)13-s + (−0.0481 − 0.132i)14-s + (−0.220 + 0.438i)15-s + (0.147 + 0.833i)16-s + (1.40 + 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.990755 - 0.0572009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990755 - 0.0572009i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.859 + 2.87i)T \) |
| good | 2 | \( 1 + (0.155 - 0.428i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (2.41 + 0.426i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (3.31 - 2.78i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (11.0 - 1.94i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-8.99 + 3.27i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-23.9 - 13.8i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (10.7 + 18.5i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-27.9 + 33.3i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (5.33 - 14.6i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-7.39 - 6.20i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-11.8 + 20.5i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-13.0 - 35.8i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (0.889 + 5.04i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-6.34 - 7.56i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 19.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (89.8 + 15.8i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-40.5 + 34.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (57.0 - 20.7i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (36.2 + 20.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-39.2 - 68.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (21.1 + 7.71i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-20.5 + 56.3i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-27.9 + 16.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (20.8 + 118. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−17.10206419246072646155826534945, −15.89779748153974201130386557825, −14.84750058651607449513847687382, −12.97678570445186603374879192837, −12.38577574055593931528138790092, −10.93135480316908667445945959174, −8.588161404755600560215447166314, −7.59659361275144242927013720161, −6.19179560246935507637032922495, −2.92890609281207700042119988620,
3.35147719157841550749868341856, 5.61430315849331084795678045261, 7.68619520966388190916284696011, 9.611722089920907163678109921315, 10.59790443167036477164762396066, 11.68310962877393418360047268186, 13.64041676007884741394197886505, 15.02030083497762687810288151736, 15.85591466785178216172409498496, 16.69895584863678804917834090555
