| L(s) = 1 | + (−50.9 + 8.98i)2-s + (58.0 + 235. i)3-s + (1.55e3 − 564. i)4-s + (1.30e3 − 1.55e3i)5-s + (−5.07e3 − 1.14e4i)6-s + (599. + 218. i)7-s + (−2.80e4 + 1.62e4i)8-s + (−5.23e4 + 2.73e4i)9-s + (−5.25e4 + 9.10e4i)10-s + (−2.71e3 − 3.23e3i)11-s + (2.23e5 + 3.33e5i)12-s + (−7.52e4 + 4.26e5i)13-s + (−3.24e4 − 5.73e3i)14-s + (4.43e5 + 2.17e5i)15-s + (−1.02e4 + 8.59e3i)16-s + (−1.61e6 − 9.29e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 + 0.280i)2-s + (0.238 + 0.971i)3-s + (1.51 − 0.551i)4-s + (0.417 − 0.498i)5-s + (−0.652 − 1.47i)6-s + (0.0356 + 0.0129i)7-s + (−0.857 + 0.494i)8-s + (−0.885 + 0.463i)9-s + (−0.525 + 0.910i)10-s + (−0.0168 − 0.0200i)11-s + (0.897 + 1.33i)12-s + (−0.202 + 1.14i)13-s + (−0.0604 − 0.0106i)14-s + (0.583 + 0.286i)15-s + (−0.00977 + 0.00819i)16-s + (−1.13 − 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0907 + 0.995i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0907 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.134974 - 0.147839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134974 - 0.147839i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-58.0 - 235. i)T \) |
| good | 2 | \( 1 + (50.9 - 8.98i)T + (962. - 350. i)T^{2} \) |
| 5 | \( 1 + (-1.30e3 + 1.55e3i)T + (-1.69e6 - 9.61e6i)T^{2} \) |
| 7 | \( 1 + (-599. - 218. i)T + (2.16e8 + 1.81e8i)T^{2} \) |
| 11 | \( 1 + (2.71e3 + 3.23e3i)T + (-4.50e9 + 2.55e10i)T^{2} \) |
| 13 | \( 1 + (7.52e4 - 4.26e5i)T + (-1.29e11 - 4.71e10i)T^{2} \) |
| 17 | \( 1 + (1.61e6 + 9.29e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (7.79e5 + 1.35e6i)T + (-3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (2.87e6 + 7.88e6i)T + (-3.17e13 + 2.66e13i)T^{2} \) |
| 29 | \( 1 + (-1.50e7 + 2.65e6i)T + (3.95e14 - 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-3.03e6 + 1.10e6i)T + (6.27e14 - 5.26e14i)T^{2} \) |
| 37 | \( 1 + (8.62e6 - 1.49e7i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + (1.09e8 + 1.92e7i)T + (1.26e16 + 4.59e15i)T^{2} \) |
| 43 | \( 1 + (-2.53e7 + 2.12e7i)T + (3.75e15 - 2.12e16i)T^{2} \) |
| 47 | \( 1 + (-9.36e7 + 2.57e8i)T + (-4.02e16 - 3.38e16i)T^{2} \) |
| 53 | \( 1 + 6.69e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (8.22e8 - 9.80e8i)T + (-8.87e16 - 5.03e17i)T^{2} \) |
| 61 | \( 1 + (-3.54e8 - 1.29e8i)T + (5.46e17 + 4.58e17i)T^{2} \) |
| 67 | \( 1 + (-2.70e8 + 1.53e9i)T + (-1.71e18 - 6.23e17i)T^{2} \) |
| 71 | \( 1 + (-2.01e8 - 1.16e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.68e9 + 2.92e9i)T + (-2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-1.65e8 - 9.37e8i)T + (-8.89e18 + 3.23e18i)T^{2} \) |
| 83 | \( 1 + (2.28e9 - 4.02e8i)T + (1.45e19 - 5.30e18i)T^{2} \) |
| 89 | \( 1 + (2.50e9 - 1.44e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (1.04e10 - 8.75e9i)T + (1.28e19 - 7.26e19i)T^{2} \) |
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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
−15.21485528575866980853258120914, −13.68486407225791464590969855155, −11.48584668919865064618738849258, −10.27425680248289040913445060482, −9.202856321056969298635920044912, −8.542162587150792995232283101823, −6.71031861779742984632649314644, −4.65388611331854496839841823608, −2.12978897906870548758098013124, −0.12353374092410700121316670661,
1.44151895040156565894597337049, 2.69514979772678581229912567261, 6.25928107373522911213783311814, 7.60128961514794660490074104043, 8.593544542363479946866095146538, 10.02695649571393371463034555192, 11.14409248090123331846098889803, 12.60081383342298036157534112767, 14.02471775387088411168738548504, 15.56971798308014056379968334200
