| L(s) = 1 | + (61.8 − 10.9i)2-s + (152. − 189. i)3-s + (2.74e3 − 1.00e3i)4-s + (−520. + 619. i)5-s + (7.35e3 − 1.33e4i)6-s + (−1.93e4 − 7.05e3i)7-s + (1.03e5 − 5.97e4i)8-s + (−1.27e4 − 5.76e4i)9-s + (−2.54e4 + 4.40e4i)10-s + (1.19e5 + 1.42e5i)11-s + (2.28e5 − 6.72e5i)12-s + (−9.05e4 + 5.13e5i)13-s + (−1.27e6 − 2.25e5i)14-s + (3.82e4 + 1.92e5i)15-s + (3.45e6 − 2.89e6i)16-s + (−3.34e5 − 1.93e5i)17-s + ⋯ |
| L(s) = 1 | + (1.93 − 0.340i)2-s + (0.626 − 0.779i)3-s + (2.68 − 0.976i)4-s + (−0.166 + 0.198i)5-s + (0.945 − 1.72i)6-s + (−1.15 − 0.420i)7-s + (3.15 − 1.82i)8-s + (−0.215 − 0.976i)9-s + (−0.254 + 0.440i)10-s + (0.744 + 0.887i)11-s + (0.919 − 2.70i)12-s + (−0.243 + 1.38i)13-s + (−2.37 − 0.418i)14-s + (0.0503 + 0.253i)15-s + (3.29 − 2.76i)16-s + (−0.235 − 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(5.34426 - 3.95441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.34426 - 3.95441i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-152. + 189. i)T \) |
| good | 2 | \( 1 + (-61.8 + 10.9i)T + (962. - 350. i)T^{2} \) |
| 5 | \( 1 + (520. - 619. i)T + (-1.69e6 - 9.61e6i)T^{2} \) |
| 7 | \( 1 + (1.93e4 + 7.05e3i)T + (2.16e8 + 1.81e8i)T^{2} \) |
| 11 | \( 1 + (-1.19e5 - 1.42e5i)T + (-4.50e9 + 2.55e10i)T^{2} \) |
| 13 | \( 1 + (9.05e4 - 5.13e5i)T + (-1.29e11 - 4.71e10i)T^{2} \) |
| 17 | \( 1 + (3.34e5 + 1.93e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-2.71e5 - 4.69e5i)T + (-3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (7.62e5 + 2.09e6i)T + (-3.17e13 + 2.66e13i)T^{2} \) |
| 29 | \( 1 + (-2.50e7 + 4.41e6i)T + (3.95e14 - 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-6.91e6 + 2.51e6i)T + (6.27e14 - 5.26e14i)T^{2} \) |
| 37 | \( 1 + (2.33e4 - 4.04e4i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + (1.58e8 + 2.78e7i)T + (1.26e16 + 4.59e15i)T^{2} \) |
| 43 | \( 1 + (1.32e8 - 1.11e8i)T + (3.75e15 - 2.12e16i)T^{2} \) |
| 47 | \( 1 + (1.96e7 - 5.40e7i)T + (-4.02e16 - 3.38e16i)T^{2} \) |
| 53 | \( 1 - 3.11e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-2.06e8 + 2.46e8i)T + (-8.87e16 - 5.03e17i)T^{2} \) |
| 61 | \( 1 + (5.76e8 + 2.09e8i)T + (5.46e17 + 4.58e17i)T^{2} \) |
| 67 | \( 1 + (8.75e7 - 4.96e8i)T + (-1.71e18 - 6.23e17i)T^{2} \) |
| 71 | \( 1 + (-1.10e9 - 6.37e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-5.27e8 - 9.13e8i)T + (-2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (8.26e8 + 4.68e9i)T + (-8.89e18 + 3.23e18i)T^{2} \) |
| 83 | \( 1 + (1.63e8 - 2.88e7i)T + (1.45e19 - 5.30e18i)T^{2} \) |
| 89 | \( 1 + (5.06e9 - 2.92e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-3.29e9 + 2.76e9i)T + (1.28e19 - 7.26e19i)T^{2} \) |
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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
−14.33526008964988445197202806815, −13.53343371512156936641708426301, −12.49124683483179995112395838192, −11.66375070300493137591263484251, −9.794419658429970184553359440574, −7.01353204847762913158576497862, −6.51973322458366904130554813816, −4.28568965745193344810466387910, −3.09323327389526904880435449273, −1.67264585794669766171734501788,
2.81692974036172755184451222204, 3.63436544623506613982108671603, 5.15951820203367811701333989461, 6.45653178676340294423681381439, 8.304807799610204337054035138199, 10.37819898662593656384852267453, 11.91972552103795687085416494143, 13.08503603557174683180872145367, 14.01287166460576062858733355638, 15.22580774704518675511441030950
