| L(s) = 1 | + (1.20 − 1.59i)2-s + (−1.09 − 3.84i)4-s + (−1.50 + 8.54i)5-s + (−7.53 − 8.98i)7-s + (−7.46 − 2.87i)8-s + (11.8 + 12.6i)10-s + (−2.27 + 0.400i)11-s + (−4.42 + 1.61i)13-s + (−23.4 + 1.21i)14-s + (−13.5 + 8.45i)16-s + (−6.77 + 11.7i)17-s + (−13.9 + 8.02i)19-s + (34.5 − 3.59i)20-s + (−2.09 + 4.10i)22-s + (−3.20 + 3.82i)23-s + ⋯ |
| L(s) = 1 | + (0.602 − 0.798i)2-s + (−0.274 − 0.961i)4-s + (−0.301 + 1.70i)5-s + (−1.07 − 1.28i)7-s + (−0.933 − 0.359i)8-s + (1.18 + 1.26i)10-s + (−0.206 + 0.0364i)11-s + (−0.340 + 0.123i)13-s + (−1.67 + 0.0870i)14-s + (−0.848 + 0.528i)16-s + (−0.398 + 0.690i)17-s + (−0.731 + 0.422i)19-s + (1.72 − 0.179i)20-s + (−0.0952 + 0.186i)22-s + (−0.139 + 0.166i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00567275 + 0.0125839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00567275 + 0.0125839i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.20 + 1.59i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.50 - 8.54i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (7.53 + 8.98i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (2.27 - 0.400i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (4.42 - 1.61i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (6.77 - 11.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (13.9 - 8.02i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (3.20 - 3.82i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (34.0 + 12.4i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-0.283 + 0.337i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-18.9 + 32.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (2.35 - 0.858i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-17.5 + 3.10i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-5.64 - 6.72i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 41.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-25.5 - 4.51i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-51.1 + 42.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (32.3 + 88.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (100. + 57.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-22.5 - 39.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-19.8 + 54.4i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (22.8 - 62.8i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (54.3 + 94.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-30.2 - 171. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.54155475294212484941400354235, −10.65505704766703882098797739064, −10.36727399878957676311195360568, −9.429734198189954687225921138613, −7.61737846422771259939367230327, −6.73760793763808855528509276498, −6.00871930688683235916807102644, −4.09352344043164249339243548893, −3.52837308354960059424717634937, −2.32441225747824089906065994241,
0.00493769910495281271527093309, 2.64243824794894460871122096264, 4.15368326423750514019371489698, 5.18109166025892603660067424858, 5.86143273230366667946370677691, 7.09219951086674137034075294246, 8.377564670357213404808067769565, 8.915926056901704140814919265778, 9.632422054343816259500752873553, 11.60452880598181668172839537533
