| L(s) = 1 | + (−0.0793 + 1.41i)2-s + (−1.98 − 0.224i)4-s + (−0.470 + 1.29i)5-s + (−1.57 − 0.277i)7-s + (0.474 − 2.78i)8-s + (−1.78 − 0.766i)10-s + (−3.66 + 1.33i)11-s + (−5.10 + 4.28i)13-s + (0.516 − 2.20i)14-s + (3.89 + 0.891i)16-s + (2.32 − 1.34i)17-s + (−3.15 − 1.82i)19-s + (1.22 − 2.46i)20-s + (−1.59 − 5.28i)22-s + (−0.644 − 3.65i)23-s + ⋯ |
| L(s) = 1 | + (−0.0561 + 0.998i)2-s + (−0.993 − 0.112i)4-s + (−0.210 + 0.577i)5-s + (−0.595 − 0.104i)7-s + (0.167 − 0.985i)8-s + (−0.564 − 0.242i)10-s + (−1.10 + 0.402i)11-s + (−1.41 + 1.18i)13-s + (0.138 − 0.588i)14-s + (0.974 + 0.222i)16-s + (0.564 − 0.326i)17-s + (−0.724 − 0.418i)19-s + (0.273 − 0.550i)20-s + (−0.339 − 1.12i)22-s + (−0.134 − 0.761i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.110151 - 0.422747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.110151 - 0.422747i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0793 - 1.41i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.470 - 1.29i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.57 + 0.277i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.66 - 1.33i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (5.10 - 4.28i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.32 + 1.34i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.15 + 1.82i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.644 + 3.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.98 - 2.36i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.45 - 0.255i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.27 - 2.20i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.30 - 7.51i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.27 + 3.49i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.901 - 5.11i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 3.96iT - 53T^{2} \) |
| 59 | \( 1 + (0.666 + 0.242i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.969 + 5.49i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 12.4i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.76 - 13.4i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.04 + 7.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.37 - 9.98i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.344 - 0.289i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.994 + 0.574i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.00 - 3.27i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.47578981616559616589349375442, −11.08583246678961349491917515904, −9.994143460148258412546478933433, −9.413137505029974972753638600630, −8.147131614761616963763740446985, −7.16317039605874783427073343538, −6.67764053800336913446494081741, −5.27558827134841724349981666073, −4.31906671329911136359520225811, −2.73797611357322218944454320482,
0.30244846774033766679027983510, 2.41677666041534383299919610389, 3.54194846429977494093653435687, 4.93363470421552995562378114111, 5.73359076879674018874977764671, 7.65294669906643524294527997123, 8.289923763717819009745053007240, 9.487399367353918936924366079200, 10.19125159130272926414015033464, 10.96305518959072265009955311550
