| L(s) = 1 | + (−11.1 + 6.41i)2-s + (−14.0 − 23.0i)3-s + (50.2 − 87.1i)4-s + (38.4 − 22.2i)5-s + (303. + 166. i)6-s + (−244. + 240. i)7-s + 469. i·8-s + (−334. + 647. i)9-s + (−284. + 493. i)10-s + (1.97e3 + 1.14e3i)11-s + (−2.71e3 + 63.9i)12-s + 1.63e3·13-s + (1.17e3 − 4.23e3i)14-s + (−1.05e3 − 574. i)15-s + (207. + 360. i)16-s + (−565. − 326. i)17-s + ⋯ |
| L(s) = 1 | + (−1.38 + 0.801i)2-s + (−0.520 − 0.854i)3-s + (0.785 − 1.36i)4-s + (0.307 − 0.177i)5-s + (1.40 + 0.768i)6-s + (−0.713 + 0.700i)7-s + 0.916i·8-s + (−0.458 + 0.888i)9-s + (−0.284 + 0.493i)10-s + (1.48 + 0.857i)11-s + (−1.57 + 0.0370i)12-s + 0.743·13-s + (0.429 − 1.54i)14-s + (−0.311 − 0.170i)15-s + (0.0507 + 0.0879i)16-s + (−0.115 − 0.0664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.478973 + 0.349072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.478973 + 0.349072i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (14.0 + 23.0i)T \) |
| 7 | \( 1 + (244. - 240. i)T \) |
| good | 2 | \( 1 + (11.1 - 6.41i)T + (32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (-38.4 + 22.2i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-1.97e3 - 1.14e3i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 1.63e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (565. + 326. i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.43e3 - 2.48e3i)T + (-2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-5.70e3 + 3.29e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 3.19e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (-2.34e4 + 4.06e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.28e4 - 2.22e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 9.09e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 8.77e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.40e4 + 8.12e3i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-5.49e4 - 3.17e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (5.73e4 + 3.31e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.26e5 - 3.92e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.23e4 + 3.87e4i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 1.33e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (2.59e5 - 4.49e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (7.38e4 + 1.27e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 2.81e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-4.69e5 + 2.71e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 8.95e4T + 8.32e11T^{2} \) |
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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
−17.16710459672251386841398295895, −16.38302802687969140590795157182, −14.93786509509914514871475921739, −13.07113665770811319733658852859, −11.64508624518045331172696135981, −9.784120311162199330505210301225, −8.649087040599358057492542455294, −6.99047306310285389375177778975, −6.03476883729969580697513278244, −1.36050624872566021364316224502,
0.72807799989705585206589816515, 3.55970826715938248186539746886, 6.44532004197380059715704049037, 8.784146687867840451423438590822, 9.804637407353480545905202009510, 10.84126965641598916967298422045, 11.84839903814584014882543254496, 13.95085215455330499168658957076, 15.95054776428716972055779621969, 16.93645967590146744694050754800
