| L(s) = 1 | + (0.173 − 0.984i)2-s + (−2.38 − 1.99i)3-s + (−0.939 − 0.342i)4-s + (−2.75 + 1.00i)5-s + (−2.38 + 1.99i)6-s + (−2.12 − 3.68i)7-s + (−0.5 + 0.866i)8-s + (1.15 + 6.57i)9-s + (0.508 + 2.88i)10-s + (0.5 − 0.866i)11-s + (1.55 + 2.69i)12-s + (3.08 − 2.58i)13-s + (−4.00 + 1.45i)14-s + (8.56 + 3.11i)15-s + (0.766 + 0.642i)16-s + (−0.539 + 3.05i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−1.37 − 1.15i)3-s + (−0.469 − 0.171i)4-s + (−1.23 + 0.448i)5-s + (−0.972 + 0.816i)6-s + (−0.804 − 1.39i)7-s + (−0.176 + 0.306i)8-s + (0.386 + 2.19i)9-s + (0.160 + 0.912i)10-s + (0.150 − 0.261i)11-s + (0.449 + 0.777i)12-s + (0.854 − 0.717i)13-s + (−1.06 + 0.389i)14-s + (2.21 + 0.805i)15-s + (0.191 + 0.160i)16-s + (−0.130 + 0.741i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00919006 + 0.00419858i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00919006 + 0.00419858i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4.29 - 0.755i)T \) |
| good | 3 | \( 1 + (2.38 + 1.99i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (2.75 - 1.00i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.12 + 3.68i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (-3.08 + 2.58i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.539 - 3.05i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (8.11 + 2.95i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.331 - 1.88i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.833 - 1.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 + (-3.59 - 3.01i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.84 - 1.03i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.02 - 5.78i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.138 + 0.0505i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.747 + 4.23i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.07 - 2.57i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.377 - 2.14i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (12.9 - 4.71i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (8.27 + 6.94i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.706 + 0.592i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.342 + 0.593i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (10.3 - 8.66i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.91 + 16.5i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.65875744038651226169202775051, −10.18896130330755820647728969076, −8.223153950771387653889090610904, −7.49954981398077341744774868873, −6.63119956069201951939533221033, −5.79127688736462495061082799228, −4.24914640251436560652415338974, −3.36202708215294002794315052669, −1.17992414749067295567239359162, −0.008791289451667381442835555176,
3.56670831767285101032047615503, 4.35500535178622275383213286858, 5.41146368565769191103628420574, 6.03738521326062832906224949470, 7.09919417061412595760273259596, 8.510880026350858258659125666331, 9.294688270147650668464822754854, 10.00455712678787447791326864642, 11.46471171442425853413584568362, 11.85644851312393122938691807055
