| L(s) = 1 | + (−1.88 − 2.10i)2-s + (−4.43 + 2.71i)3-s + (−0.902 + 7.94i)4-s + (5.32 − 14.6i)5-s + (14.0 + 4.24i)6-s + (6.67 + 1.17i)7-s + (18.4 − 13.0i)8-s + (12.3 − 24.0i)9-s + (−40.9 + 16.3i)10-s + (−32.8 + 11.9i)11-s + (−17.5 − 37.6i)12-s + (−58.0 + 48.7i)13-s + (−10.0 − 16.3i)14-s + (16.0 + 79.3i)15-s + (−62.3 − 14.3i)16-s + (−26.5 + 15.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.666 − 0.745i)2-s + (−0.853 + 0.521i)3-s + (−0.112 + 0.993i)4-s + (0.476 − 1.30i)5-s + (0.957 + 0.288i)6-s + (0.360 + 0.0635i)7-s + (0.816 − 0.577i)8-s + (0.455 − 0.890i)9-s + (−1.29 + 0.516i)10-s + (−0.900 + 0.327i)11-s + (−0.422 − 0.906i)12-s + (−1.23 + 1.04i)13-s + (−0.192 − 0.311i)14-s + (0.276 + 1.36i)15-s + (−0.974 − 0.224i)16-s + (−0.378 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0195386 + 0.0610780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0195386 + 0.0610780i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.88 + 2.10i)T \) |
| 3 | \( 1 + (4.43 - 2.71i)T \) |
| good | 5 | \( 1 + (-5.32 + 14.6i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-6.67 - 1.17i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (32.8 - 11.9i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (58.0 - 48.7i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (26.5 - 15.2i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (47.8 + 27.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (16.2 + 91.9i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (146. - 174. i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (134. - 23.6i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (109. + 189. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-42.2 - 50.2i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-37.6 - 103. i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (1.93 - 10.9i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 586. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-380. - 138. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (122. - 695. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-518. - 617. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (526. + 911. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-335. + 581. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (715. - 852. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (539. + 452. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-731. - 422. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-930. + 338. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.53412358651778595795238603822, −11.46665543858666905398618214701, −10.43010966611911704496107375487, −9.457090453474381484993699003875, −8.682358539980900092511834554028, −7.10136388566883881584268567081, −5.18636421401821871815613087944, −4.38360884428628799143546408506, −1.91192421603864829817263100612, −0.04573571756737732110792078637,
2.25328268417482511565483926110, 5.18934007961421201879249057952, 6.08713990629440379408643494299, 7.25923165936804029479032571258, 7.904636125631270529559923481752, 9.857998334472726334291966596513, 10.57497022550822413994295246204, 11.35026029250082266311875960489, 12.96898355600520055232352222158, 13.99125498050781951124238526599
