| L(s) = 1 | + (−4.77 − 1.73i)2-s + (11.5 + 10.4i)3-s + (−4.74 − 3.98i)4-s + (−2.77 + 15.7i)5-s + (−37.2 − 69.9i)6-s + (−95.3 + 80.0i)7-s + (97.0 + 168. i)8-s + (25.6 + 241. i)9-s + (40.6 − 70.4i)10-s + (98.9 + 561. i)11-s + (−13.4 − 95.6i)12-s + (−475. + 173. i)13-s + (594. − 216. i)14-s + (−196. + 153. i)15-s + (−136. − 775. i)16-s + (401. − 694. i)17-s + ⋯ |
| L(s) = 1 | + (−0.843 − 0.307i)2-s + (0.743 + 0.668i)3-s + (−0.148 − 0.124i)4-s + (−0.0497 + 0.281i)5-s + (−0.421 − 0.792i)6-s + (−0.735 + 0.617i)7-s + (0.535 + 0.928i)8-s + (0.105 + 0.994i)9-s + (0.128 − 0.222i)10-s + (0.246 + 1.39i)11-s + (−0.0270 − 0.191i)12-s + (−0.780 + 0.284i)13-s + (0.810 − 0.294i)14-s + (−0.225 + 0.176i)15-s + (−0.133 − 0.757i)16-s + (0.336 − 0.583i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.553866 + 0.630654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.553866 + 0.630654i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-11.5 - 10.4i)T \) |
| good | 2 | \( 1 + (4.77 + 1.73i)T + (24.5 + 20.5i)T^{2} \) |
| 5 | \( 1 + (2.77 - 15.7i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (95.3 - 80.0i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-98.9 - 561. i)T + (-1.51e5 + 5.50e4i)T^{2} \) |
| 13 | \( 1 + (475. - 173. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-401. + 694. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (992. + 1.71e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.01e3 - 1.68e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (2.69e3 + 979. i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (813. + 682. i)T + (4.97e6 + 2.81e7i)T^{2} \) |
| 37 | \( 1 + (-5.03e3 + 8.71e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-7.76e3 + 2.82e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-2.89e3 - 1.64e4i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (1.34e4 - 1.12e4i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 - 1.50e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.36e3 + 7.73e3i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (2.61e4 - 2.19e4i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-5.84e4 + 2.12e4i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (5.19e3 - 8.99e3i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.11e4 + 1.92e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-8.04e4 - 2.92e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-3.07e4 - 1.11e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-5.81e4 - 1.00e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (2.83e4 + 1.60e5i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−16.74291752088500216521573154089, −15.27147541008276191886413470617, −14.48308885935244906649310378190, −12.87839335863588064428007312170, −11.04201640139850376083133279265, −9.568287236328118290876646560527, −9.262317595499749500665244801969, −7.40667475755994878858925827052, −4.80528912901942354893790370237, −2.46916590646894763441411178285,
0.68512400005782234807454105862, 3.56464007544064615681323926236, 6.60960640730547916426476770339, 7.987298535716976726420837398228, 8.930062905846852881976720828552, 10.28438471328133353206618603550, 12.52136335959376881533333659756, 13.35978402375658100872686141464, 14.66946239707633602663523259257, 16.46274839958376578192885944460
