| L(s) = 1 | + (−1.16 − 1.60i)2-s + (−4.81 − 14.8i)3-s + (18.5 − 57.1i)4-s + (−1.49 − 1.08i)5-s + (−18.2 + 25.0i)6-s + (−25.7 − 8.37i)7-s + (−234. + 76.2i)8-s + (−196. + 142. i)9-s + 3.68i·10-s + (−1.31e3 − 220. i)11-s − 935.·12-s + (−1.02e3 − 1.40e3i)13-s + (16.6 + 51.3i)14-s + (−8.91 + 27.4i)15-s + (−2.71e3 − 1.97e3i)16-s + (−1.48e3 + 2.04e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.146 − 0.201i)2-s + (−0.178 − 0.549i)3-s + (0.289 − 0.892i)4-s + (−0.0119 − 0.00870i)5-s + (−0.0843 + 0.116i)6-s + (−0.0751 − 0.0244i)7-s + (−0.458 + 0.148i)8-s + (−0.269 + 0.195i)9-s + 0.00368i·10-s + (−0.986 − 0.165i)11-s − 0.541·12-s + (−0.465 − 0.640i)13-s + (0.00607 + 0.0186i)14-s + (−0.00264 + 0.00813i)15-s + (−0.662 − 0.480i)16-s + (−0.302 + 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.100458 - 0.921460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100458 - 0.921460i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.81 + 14.8i)T \) |
| 11 | \( 1 + (1.31e3 + 220. i)T \) |
| good | 2 | \( 1 + (1.16 + 1.60i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (1.49 + 1.08i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (25.7 + 8.37i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (1.02e3 + 1.40e3i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (1.48e3 - 2.04e3i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (1.80e3 - 586. i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 - 9.11e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-2.49e4 - 8.11e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (-2.89e4 + 2.10e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (-1.52e4 + 4.70e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (3.59e4 - 1.16e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 + 7.66e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (1.86e4 + 5.73e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (7.85e3 - 5.70e3i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (-3.16e4 + 9.72e4i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-5.68e4 + 7.82e4i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 - 1.08e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-2.72e5 - 1.98e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-3.21e5 - 1.04e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-3.81e5 - 5.25e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-5.74e5 + 7.90e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 + 6.44e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (1.02e6 - 7.42e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−14.91409083826568216163716035491, −13.59851328832054650701407010622, −12.36331994235297287836874660913, −10.97372859622106236913120651673, −10.00657445987451266846832964537, −8.224261146433159513907967228290, −6.59678950813388689872663743991, −5.23670711454308911937148192145, −2.41846846497529298857677390856, −0.49411640896590201097212146239,
2.87495031513459003555753749901, 4.73025721026008670120128688845, 6.73429087622700340643294019642, 8.151225611471758260097254201507, 9.553422251736374699027641459811, 11.06393879026333454014230303794, 12.21978629669509108450777730771, 13.48713034862153650185480223991, 15.16828217821923258384433190521, 16.02605030628965199545370724917
