| L(s) = 1 | + (−1.39 − 0.245i)2-s + (−2.10 + 2.13i)3-s + (1.87 + 0.684i)4-s + (−5.37 − 6.40i)5-s + (3.45 − 2.45i)6-s + (−8.61 + 3.13i)7-s + (−2.44 − 1.41i)8-s + (−0.136 − 8.99i)9-s + (5.91 + 10.2i)10-s + (−4.22 + 5.04i)11-s + (−5.41 + 2.57i)12-s + (1.04 + 5.91i)13-s + (12.7 − 2.25i)14-s + (25.0 + 1.99i)15-s + (3.06 + 2.57i)16-s + (−0.880 + 0.508i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (−0.701 + 0.712i)3-s + (0.469 + 0.171i)4-s + (−1.07 − 1.28i)5-s + (0.576 − 0.409i)6-s + (−1.23 + 0.448i)7-s + (−0.306 − 0.176i)8-s + (−0.0151 − 0.999i)9-s + (0.591 + 1.02i)10-s + (−0.384 + 0.458i)11-s + (−0.451 + 0.214i)12-s + (0.0802 + 0.455i)13-s + (0.912 − 0.160i)14-s + (1.66 + 0.133i)15-s + (0.191 + 0.160i)16-s + (−0.0517 + 0.0298i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00539746 - 0.0474864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00539746 - 0.0474864i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.245i)T \) |
| 3 | \( 1 + (2.10 - 2.13i)T \) |
| good | 5 | \( 1 + (5.37 + 6.40i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (8.61 - 3.13i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (4.22 - 5.04i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (-1.04 - 5.91i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (0.880 - 0.508i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-10.5 + 18.2i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (7.33 - 20.1i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (47.7 + 8.41i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (6.20 + 2.25i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (33.2 + 57.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-52.2 + 9.20i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (36.3 + 30.5i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (2.23 + 6.13i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 - 39.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (18.0 + 21.4i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-38.6 + 14.0i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (2.57 + 14.6i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (65.6 - 37.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (26.5 - 45.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.2 - 80.6i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-46.2 - 8.14i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-20.2 - 11.7i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-26.8 - 22.4i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−15.39272962771982961042377731313, −12.93641441191020690626029479923, −12.12686911878517533131974200275, −11.17620411396599796302021988166, −9.579009143745517561189681981839, −8.991164600612885149918788491906, −7.28546357094058625613403430664, −5.48636438132931184867580122558, −3.85005782524118181190500196780, −0.06063835335364426742464483240,
3.24262616861434414148522131745, 6.12399087120482399586611543335, 7.11019968248089393826613717908, 7.999414546603424601404284804740, 10.13497339378946540719101114062, 10.92139577924473397736773857439, 11.97956887729132234336373694856, 13.22907657519646971922237598288, 14.70035817732095885545443332901, 16.00697145207135820479350126338
