| L(s) = 1 | + (59.8 − 10.5i)2-s + (−242. + 15.1i)3-s + (2.50e3 − 910. i)4-s + (−353. + 421. i)5-s + (−1.43e4 + 3.46e3i)6-s + (1.98e4 + 7.21e3i)7-s + (8.62e4 − 4.97e4i)8-s + (5.85e4 − 7.35e3i)9-s + (−1.66e4 + 2.89e4i)10-s + (−4.92e4 − 5.86e4i)11-s + (−5.93e5 + 2.58e5i)12-s + (1.02e5 − 5.83e5i)13-s + (1.26e6 + 2.22e5i)14-s + (7.93e4 − 1.07e5i)15-s + (2.54e6 − 2.13e6i)16-s + (1.23e6 + 7.14e5i)17-s + ⋯ |
| L(s) = 1 | + (1.86 − 0.329i)2-s + (−0.998 + 0.0624i)3-s + (2.44 − 0.889i)4-s + (−0.113 + 0.134i)5-s + (−1.84 + 0.445i)6-s + (1.17 + 0.429i)7-s + (2.63 − 1.51i)8-s + (0.992 − 0.124i)9-s + (−0.166 + 0.289i)10-s + (−0.305 − 0.364i)11-s + (−2.38 + 1.04i)12-s + (0.276 − 1.57i)13-s + (2.34 + 0.413i)14-s + (0.104 − 0.141i)15-s + (2.42 − 2.03i)16-s + (0.871 + 0.503i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(4.58551 - 1.48686i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.58551 - 1.48686i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (242. - 15.1i)T \) |
| good | 2 | \( 1 + (-59.8 + 10.5i)T + (962. - 350. i)T^{2} \) |
| 5 | \( 1 + (353. - 421. i)T + (-1.69e6 - 9.61e6i)T^{2} \) |
| 7 | \( 1 + (-1.98e4 - 7.21e3i)T + (2.16e8 + 1.81e8i)T^{2} \) |
| 11 | \( 1 + (4.92e4 + 5.86e4i)T + (-4.50e9 + 2.55e10i)T^{2} \) |
| 13 | \( 1 + (-1.02e5 + 5.83e5i)T + (-1.29e11 - 4.71e10i)T^{2} \) |
| 17 | \( 1 + (-1.23e6 - 7.14e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (9.04e5 + 1.56e6i)T + (-3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-3.09e6 - 8.49e6i)T + (-3.17e13 + 2.66e13i)T^{2} \) |
| 29 | \( 1 + (7.59e6 - 1.33e6i)T + (3.95e14 - 1.43e14i)T^{2} \) |
| 31 | \( 1 + (3.51e7 - 1.28e7i)T + (6.27e14 - 5.26e14i)T^{2} \) |
| 37 | \( 1 + (-8.13e6 + 1.40e7i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + (7.58e7 + 1.33e7i)T + (1.26e16 + 4.59e15i)T^{2} \) |
| 43 | \( 1 + (6.39e7 - 5.36e7i)T + (3.75e15 - 2.12e16i)T^{2} \) |
| 47 | \( 1 + (-1.19e8 + 3.28e8i)T + (-4.02e16 - 3.38e16i)T^{2} \) |
| 53 | \( 1 - 4.58e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (3.90e8 - 4.65e8i)T + (-8.87e16 - 5.03e17i)T^{2} \) |
| 61 | \( 1 + (1.42e9 + 5.20e8i)T + (5.46e17 + 4.58e17i)T^{2} \) |
| 67 | \( 1 + (-1.08e8 + 6.15e8i)T + (-1.71e18 - 6.23e17i)T^{2} \) |
| 71 | \( 1 + (-2.23e8 - 1.29e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.53e9 - 2.66e9i)T + (-2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-3.06e7 - 1.73e8i)T + (-8.89e18 + 3.23e18i)T^{2} \) |
| 83 | \( 1 + (2.01e9 - 3.56e8i)T + (1.45e19 - 5.30e18i)T^{2} \) |
| 89 | \( 1 + (-3.29e9 + 1.90e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (1.52e8 - 1.28e8i)T + (1.28e19 - 7.26e19i)T^{2} \) |
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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
−14.97709183892771201958523544535, −13.38141470359493915390191363435, −12.42461382264651293024658887712, −11.28724789286923925129601335113, −10.66196313680774662181054001978, −7.47243574507211097118351948728, −5.66896550824972389625689975824, −5.14913091006635160281929021222, −3.42878974859698842678023448409, −1.45804615670236188409438335656,
1.77708656651421866301165288962, 4.20723761500764101279521529216, 4.97905805545868692337265930050, 6.41160450237688931648426394342, 7.60817914647181665896529104998, 10.76719365500154767760869811737, 11.71082438052019528815632411459, 12.57975412141689474299772409147, 13.96671354638129882320137326096, 14.81593260932137978042453798028
