| L(s) = 1 | + (−0.00307 + 1.73i)3-s + (2.47 + 0.901i)5-s + (−2.55 − 0.451i)7-s + (−2.99 − 0.0106i)9-s + (−0.556 − 1.52i)11-s + (1.88 + 2.24i)13-s + (−1.56 + 4.28i)15-s + (−3.28 + 1.89i)17-s + (−4.30 + 7.46i)19-s + (0.789 − 4.43i)21-s + (1.07 + 6.06i)23-s + (1.49 + 1.25i)25-s + (0.0276 − 5.19i)27-s + (3.88 + 3.25i)29-s + (3.57 − 0.630i)31-s + ⋯ |
| L(s) = 1 | + (−0.00177 + 0.999i)3-s + (1.10 + 0.403i)5-s + (−0.967 − 0.170i)7-s + (−0.999 − 0.00355i)9-s + (−0.167 − 0.461i)11-s + (0.522 + 0.622i)13-s + (−0.405 + 1.10i)15-s + (−0.797 + 0.460i)17-s + (−0.988 + 1.71i)19-s + (0.172 − 0.967i)21-s + (0.223 + 1.26i)23-s + (0.298 + 0.250i)25-s + (0.00533 − 0.999i)27-s + (0.720 + 0.604i)29-s + (0.641 − 0.113i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.420431 + 1.18498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.420431 + 1.18498i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.00307 - 1.73i)T \) |
| good | 5 | \( 1 + (-2.47 - 0.901i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (2.55 + 0.451i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.556 + 1.52i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.88 - 2.24i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.28 - 1.89i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.30 - 7.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.07 - 6.06i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.88 - 3.25i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.57 + 0.630i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (6.40 - 3.69i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.43 + 5.28i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.77 + 1.37i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.253 - 1.43i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.180T + 53T^{2} \) |
| 59 | \( 1 + (0.253 - 0.695i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-11.1 - 1.96i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.5 + 8.82i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.57 + 4.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.62 - 4.55i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 + 8.01i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.39 - 8.81i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.211 + 0.121i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.95 + 0.711i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.31291560697237310781978067929, −9.848399427947023919389913827322, −8.983951397868802562331148593774, −8.278462614051096464949849169698, −6.66265322892190356023217177435, −6.16625032930011117442285431792, −5.35387201607269256015278258106, −4.00450925397566415239560490378, −3.27763498282461510262546200519, −1.97530596800559512929374155116,
0.57080271049951045670750451152, 2.18489504149867796500822595910, 2.85945685449788576878848821458, 4.63628918362869352556947330547, 5.64049086895031377750230081834, 6.58387330657522810771244385502, 6.85006694532589480915704037317, 8.349840370441068043249813186517, 8.886598530538144634468561575438, 9.745680230369407971037224677588
