| L(s) = 1 | + (−0.891 + 0.238i)2-s + (−2.93 − 0.613i)3-s + (−2.72 + 1.57i)4-s + (−1.31 − 4.82i)5-s + (2.76 − 0.154i)6-s + (−11.0 + 2.96i)7-s + (4.66 − 4.66i)8-s + (8.24 + 3.60i)9-s + (2.32 + 3.98i)10-s + (−1.30 + 2.26i)11-s + (8.97 − 2.94i)12-s + (2.85 + 0.764i)13-s + (9.15 − 5.28i)14-s + (0.908 + 14.9i)15-s + (3.24 − 5.62i)16-s + (−13.6 − 13.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.445 + 0.119i)2-s + (−0.978 − 0.204i)3-s + (−0.681 + 0.393i)4-s + (−0.263 − 0.964i)5-s + (0.460 − 0.0257i)6-s + (−1.57 + 0.423i)7-s + (0.583 − 0.583i)8-s + (0.916 + 0.400i)9-s + (0.232 + 0.398i)10-s + (−0.118 + 0.205i)11-s + (0.747 − 0.245i)12-s + (0.219 + 0.0588i)13-s + (0.653 − 0.377i)14-s + (0.0605 + 0.998i)15-s + (0.203 − 0.351i)16-s + (−0.805 − 0.805i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00883755 - 0.0753363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00883755 - 0.0753363i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.93 + 0.613i)T \) |
| 5 | \( 1 + (1.31 + 4.82i)T \) |
| good | 2 | \( 1 + (0.891 - 0.238i)T + (3.46 - 2i)T^{2} \) |
| 7 | \( 1 + (11.0 - 2.96i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (1.30 - 2.26i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-2.85 - 0.764i)T + (146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (13.6 + 13.6i)T + 289iT^{2} \) |
| 19 | \( 1 - 11.4iT - 361T^{2} \) |
| 23 | \( 1 + (10.7 + 2.89i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (23.0 + 13.2i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (21.8 + 37.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-14.4 - 14.4i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (0.924 + 1.60i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-13.1 - 49.0i)T + (-1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (-61.3 + 16.4i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-6.43 + 6.43i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (49.5 - 28.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-16.8 + 29.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (8.41 - 31.4i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 63.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (50.9 - 50.9i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (59.5 + 34.4i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (27.2 + 101. i)T + (-5.96e3 + 3.44e3i)T^{2} \) |
| 89 | \( 1 + 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-35.3 + 9.45i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−15.74506975849466771794885377177, −13.28964233527661404395910286129, −12.83989734527563251362243609728, −11.77228005436347436422326954664, −9.922131869829286878968921186388, −9.075346681453765638725233321055, −7.48261849523490948154929272899, −5.88671328211566458818444045397, −4.23608471815439681462766068793, −0.10344797455491693729038421039,
3.88970192032924855507853808204, 5.91356579067965384402021214356, 7.07967646450504475361302308778, 9.195417246657395744438245237521, 10.36004510580348200770618132911, 10.92580357236198921807755371520, 12.65778505027315745170098916918, 13.71458643830547044720790113801, 15.24174971228115613056315835615, 16.22764442891380521289493301971
