| L(s) = 1 | + (−2.67 − 1.36i)3-s + (−3.07 + 5.32i)5-s + (−0.511 + 0.295i)7-s + (5.27 + 7.29i)9-s + (15.1 − 8.72i)11-s + (0.892 − 1.54i)13-s + (15.4 − 10.0i)15-s − 16.9·17-s + 19.5i·19-s + (1.76 − 0.0911i)21-s + (6.86 + 3.96i)23-s + (−6.39 − 11.0i)25-s + (−4.15 − 26.6i)27-s + (−3.17 − 5.49i)29-s + (−27.6 − 15.9i)31-s + ⋯ |
| L(s) = 1 | + (−0.890 − 0.454i)3-s + (−0.614 + 1.06i)5-s + (−0.0730 + 0.0421i)7-s + (0.586 + 0.810i)9-s + (1.37 − 0.793i)11-s + (0.0686 − 0.118i)13-s + (1.03 − 0.668i)15-s − 0.995·17-s + 1.02i·19-s + (0.0842 − 0.00433i)21-s + (0.298 + 0.172i)23-s + (−0.255 − 0.443i)25-s + (−0.153 − 0.988i)27-s + (−0.109 − 0.189i)29-s + (−0.892 − 0.515i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2535710149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2535710149\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.67 + 1.36i)T \) |
| good | 5 | \( 1 + (3.07 - 5.32i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (0.511 - 0.295i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.1 + 8.72i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-0.892 + 1.54i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 16.9T + 289T^{2} \) |
| 19 | \( 1 - 19.5iT - 361T^{2} \) |
| 23 | \( 1 + (-6.86 - 3.96i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (3.17 + 5.49i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (27.6 + 15.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 58.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (2.66 - 4.62i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (33.9 - 19.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-9.64 + 5.56i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 35.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (20.8 + 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.9 - 65.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (31.8 + 18.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 87.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (32.1 - 18.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (66.0 - 38.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 27.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-13.0 - 22.6i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.11066255776303382136517882138, −10.37754948334526356149725553442, −9.181960814559031725121251520290, −8.085095594874686642215251772850, −7.09052023761266716463013329550, −6.50724291931428483920375148872, −5.68309977060947660935446579046, −4.23952286934582785898535713571, −3.27628966309031088029845663802, −1.60203460419264545369124033312,
0.11435109300104871234788463649, 1.51686031109618766160823563131, 3.72292969749123659701250107911, 4.53952770170907454724405702379, 5.17776503890349094402461892774, 6.58885374135017403019265885149, 7.12420934237137098683561257538, 8.755437874618431067351603460588, 9.077900378539448686334042111550, 10.12507792701925315392070923836
