| L(s) = 1 | + (1.49 + 0.869i)3-s + (−0.259 + 0.712i)5-s + (−3.61 − 0.638i)7-s + (1.48 + 2.60i)9-s + (−1.44 + 0.527i)11-s + (−2.02 + 1.69i)13-s + (−1.00 + 0.841i)15-s + (−6.43 + 3.71i)17-s + (2.42 + 1.39i)19-s + (−4.86 − 4.10i)21-s + (−0.216 − 1.22i)23-s + (3.39 + 2.84i)25-s + (−0.0384 + 5.19i)27-s + (−0.239 + 0.284i)29-s + (−5.65 + 0.996i)31-s + ⋯ |
| L(s) = 1 | + (0.864 + 0.502i)3-s + (−0.115 + 0.318i)5-s + (−1.36 − 0.241i)7-s + (0.495 + 0.868i)9-s + (−0.436 + 0.158i)11-s + (−0.560 + 0.470i)13-s + (−0.260 + 0.217i)15-s + (−1.55 + 0.900i)17-s + (0.555 + 0.320i)19-s + (−1.06 − 0.895i)21-s + (−0.0451 − 0.256i)23-s + (0.678 + 0.568i)25-s + (−0.00740 + 0.999i)27-s + (−0.0443 + 0.0529i)29-s + (−1.01 + 0.178i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.356680 + 1.01405i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.356680 + 1.01405i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.49 - 0.869i)T \) |
| good | 5 | \( 1 + (0.259 - 0.712i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.61 + 0.638i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.44 - 0.527i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.02 - 1.69i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.43 - 3.71i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.42 - 1.39i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.216 + 1.22i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.239 - 0.284i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (5.65 - 0.996i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.19 - 3.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.76 - 6.87i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.21 + 8.83i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.299 + 1.69i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 5.98iT - 53T^{2} \) |
| 59 | \( 1 + (-2.42 - 0.881i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.58 + 8.96i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.87 + 10.5i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.47 - 11.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.46 + 7.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.83 + 8.14i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.595 - 0.500i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-13.5 - 7.83i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.19 + 1.16i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.44160913980684827348902331293, −9.501638920879119106314547054278, −9.077915524904918724815379059200, −7.988731205436878621311297109629, −7.08139071690752174473749960102, −6.39500068418842222197251091964, −5.00642684893904906324854229205, −3.97279642671355552162262312234, −3.16719980121608335587118207282, −2.14588815625528716624554679156,
0.43173426177154977100745207572, 2.40281986746132328228198347247, 3.06081887174370099851597536071, 4.25090922579001581708266992738, 5.49592196699858353450732750725, 6.62802131432529800085309099509, 7.22577774624305121496711203050, 8.168900862972913008023891416964, 9.252297723433205142036029353906, 9.377901267060186565802846137290
