| L(s) = 1 | + (−0.781 − 0.623i)2-s + (−0.473 − 1.66i)3-s + (0.222 + 0.974i)4-s + (−0.256 − 3.42i)5-s + (−0.668 + 1.59i)6-s + (−2.20 − 1.46i)7-s + (0.433 − 0.900i)8-s + (−2.55 + 1.57i)9-s + (−1.93 + 2.84i)10-s + (0.161 − 0.0634i)11-s + (1.51 − 0.832i)12-s + (−4.51 + 1.77i)13-s + (0.812 + 2.51i)14-s + (−5.58 + 2.05i)15-s + (−0.900 + 0.433i)16-s + (−0.839 + 0.258i)17-s + ⋯ |
| L(s) = 1 | + (−0.552 − 0.440i)2-s + (−0.273 − 0.961i)3-s + (0.111 + 0.487i)4-s + (−0.114 − 1.53i)5-s + (−0.272 + 0.652i)6-s + (−0.833 − 0.552i)7-s + (0.153 − 0.318i)8-s + (−0.850 + 0.526i)9-s + (−0.612 + 0.898i)10-s + (0.0487 − 0.0191i)11-s + (0.438 − 0.240i)12-s + (−1.25 + 0.491i)13-s + (0.217 + 0.672i)14-s + (−1.44 + 0.529i)15-s + (−0.225 + 0.108i)16-s + (−0.203 + 0.0627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.174342 + 0.143096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174342 + 0.143096i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.781 + 0.623i)T \) |
| 3 | \( 1 + (0.473 + 1.66i)T \) |
| 7 | \( 1 + (2.20 + 1.46i)T \) |
| good | 5 | \( 1 + (0.256 + 3.42i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-0.161 + 0.0634i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (4.51 - 1.77i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (0.839 - 0.258i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-1.69 - 0.976i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.64 - 5.01i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.96 + 6.35i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + 8.07iT - 31T^{2} \) |
| 37 | \( 1 + (-2.26 - 2.09i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-5.83 + 3.97i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (9.50 + 6.47i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (3.61 - 4.53i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-3.57 - 3.85i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-3.58 + 1.72i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 2.36i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + (11.8 - 2.71i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.843 - 0.330i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + 8.46T + 79T^{2} \) |
| 83 | \( 1 + (-0.178 + 0.455i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-3.37 + 0.508i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (1.44 - 0.835i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.426895215729539576450213974581, −8.757224302885014169883596874950, −7.68336294356457414659655821646, −7.25857496980897475715049076490, −6.05972385468140947027346783982, −5.04372086572360600655670622247, −3.96929654454259136752526602248, −2.48625074790258876260592995387, −1.23118093398733764750316571282, −0.14364453584928282629358911292,
2.72668934783820353731434084847, 3.22066542929367164959552435618, 4.78984828900722071917268615808, 5.68276410159090655652477398150, 6.73683461998804899387369507215, 7.07321930210664626102101705093, 8.420763846242351843482160834795, 9.298172046418203762434369874085, 10.01447157066977540829416906783, 10.51789439015470272119085204300
