| L(s) = 1 | + (2.69 − 3.21i)5-s + (11.1 + 4.05i)7-s + (−7.66 − 9.13i)11-s + (−0.429 + 2.43i)13-s + (11.9 + 6.92i)17-s + (1.88 + 3.27i)19-s + (−9.18 − 25.2i)23-s + (1.28 + 7.30i)25-s + (40.6 − 7.16i)29-s + (32.7 − 11.9i)31-s + (43.0 − 24.8i)35-s + (33.5 − 58.0i)37-s + (−5.01 − 0.883i)41-s + (−35.5 + 29.8i)43-s + (−31.9 + 87.6i)47-s + ⋯ |
| L(s) = 1 | + (0.539 − 0.642i)5-s + (1.59 + 0.579i)7-s + (−0.696 − 0.830i)11-s + (−0.0330 + 0.187i)13-s + (0.705 + 0.407i)17-s + (0.0994 + 0.172i)19-s + (−0.399 − 1.09i)23-s + (0.0515 + 0.292i)25-s + (1.40 − 0.246i)29-s + (1.05 − 0.385i)31-s + (1.22 − 0.709i)35-s + (0.906 − 1.56i)37-s + (−0.122 − 0.0215i)41-s + (−0.826 + 0.693i)43-s + (−0.678 + 1.86i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.05498 - 0.372755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05498 - 0.372755i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.69 + 3.21i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-11.1 - 4.05i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (7.66 + 9.13i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.429 - 2.43i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-11.9 - 6.92i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-1.88 - 3.27i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (9.18 + 25.2i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-40.6 + 7.16i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-32.7 + 11.9i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-33.5 + 58.0i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (5.01 + 0.883i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (35.5 - 29.8i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (31.9 - 87.6i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 51.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (9.45 - 11.2i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (48.1 + 17.5i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-2.35 + 13.3i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-27.9 - 16.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (20.3 + 35.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-12.9 - 73.1i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (150. - 26.6i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-18.0 + 10.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (36.5 - 30.6i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−11.34996986592039711592991409962, −10.50970678682860428927364250778, −9.389762746400082985185274498494, −8.260064251609650845168571647115, −8.018226333457355745053380738723, −6.19245114448373426679509252297, −5.32251391963744807771507211526, −4.48157813753098228649501244261, −2.59699885991430489774520996447, −1.24373942082012802175721705350,
1.47395440697324362044929695717, 2.83327987903669193818595777599, 4.53779702459759884584102212484, 5.31793461923764736093169381501, 6.73262367494155606398367914158, 7.67299181610731968813304447351, 8.385948529529188662713654818189, 10.03968873878549817378839346645, 10.28297968052208142083354404575, 11.45333415987469860975751025355
