L(s) = 1 | + (−2.81 + 0.253i)2-s + (5.49 + 5.49i)3-s + (7.87 − 1.42i)4-s + (−4.66 + 4.66i)5-s + (−16.8 − 14.0i)6-s − 24.8i·7-s + (−21.8 + 6.00i)8-s + 33.4i·9-s + (11.9 − 14.3i)10-s + (22.3 − 22.3i)11-s + (51.1 + 35.4i)12-s + (−11.2 − 11.2i)13-s + (6.30 + 70.1i)14-s − 51.2·15-s + (59.9 − 22.4i)16-s − 88.4·17-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.0894i)2-s + (1.05 + 1.05i)3-s + (0.983 − 0.178i)4-s + (−0.417 + 0.417i)5-s + (−1.14 − 0.958i)6-s − 1.34i·7-s + (−0.964 + 0.265i)8-s + 1.23i·9-s + (0.378 − 0.452i)10-s + (0.612 − 0.612i)11-s + (1.22 + 0.852i)12-s + (−0.240 − 0.240i)13-s + (0.120 + 1.33i)14-s − 0.882·15-s + (0.936 − 0.350i)16-s − 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.799302 + 0.315141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.799302 + 0.315141i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.81 - 0.253i)T \) |
good | 3 | \( 1 + (-5.49 - 5.49i)T + 27iT^{2} \) |
| 5 | \( 1 + (4.66 - 4.66i)T - 125iT^{2} \) |
| 7 | \( 1 + 24.8iT - 343T^{2} \) |
| 11 | \( 1 + (-22.3 + 22.3i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (11.2 + 11.2i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + 88.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-37.8 - 37.8i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 48.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-10.4 - 10.4i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 96.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + (163. - 163. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 360. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (100. - 100. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (175. - 175. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-405. + 405. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-664. - 664. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (107. + 107. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 215. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 668. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 822.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-326. - 326. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 262. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 150.T + 9.12e5T^{2} \) |
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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
−19.18378111044967489493928575706, −17.37080965920219412077854324230, −16.16604314025702061232247799118, −15.10379471695498216268016039356, −13.92801851059991218948762916483, −11.16800947255393671871768887243, −10.08574417320134893822762914855, −8.756413192025216190940882303818, −7.25148595476539696283598629308, −3.60641650265063205159219435507,
2.25265769513463768454853155610, 6.86678798802957610728685683430, 8.395618962321159118455322537640, 9.208322658304332383728951439167, 11.75882876176833103648751854754, 12.74861254480874746755156540661, 14.69988507753987762872501130772, 15.86252342172390217843419830064, 17.70774264941209122571400972689, 18.64532290781138716466997533540