| L(s) = 1 | + (−0.805 + 1.16i)2-s + (1.23 − 1.21i)3-s + (−0.702 − 1.87i)4-s + (−0.710 + 1.95i)5-s + (0.422 + 2.41i)6-s + (3.83 + 0.677i)7-s + (2.74 + 0.691i)8-s + (0.0366 − 2.99i)9-s + (−1.69 − 2.39i)10-s + (1.46 − 0.533i)11-s + (−3.14 − 1.45i)12-s + (−1.23 + 1.03i)13-s + (−3.87 + 3.91i)14-s + (1.50 + 3.26i)15-s + (−3.01 + 2.63i)16-s + (−3.61 + 2.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.569 + 0.821i)2-s + (0.711 − 0.702i)3-s + (−0.351 − 0.936i)4-s + (−0.317 + 0.872i)5-s + (0.172 + 0.985i)6-s + (1.45 + 0.255i)7-s + (0.969 + 0.244i)8-s + (0.0122 − 0.999i)9-s + (−0.536 − 0.758i)10-s + (0.442 − 0.160i)11-s + (−0.907 − 0.419i)12-s + (−0.341 + 0.286i)13-s + (−1.03 + 1.04i)14-s + (0.387 + 0.844i)15-s + (−0.753 + 0.657i)16-s + (−0.877 + 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.944798 + 0.263633i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.944798 + 0.263633i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.805 - 1.16i)T \) |
| 3 | \( 1 + (-1.23 + 1.21i)T \) |
| good | 5 | \( 1 + (0.710 - 1.95i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.83 - 0.677i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.46 + 0.533i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.23 - 1.03i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.61 - 2.08i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.96 + 4.02i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.659 + 3.74i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.44 - 2.91i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.621 + 0.109i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.912 - 1.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.49 + 4.15i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.14 - 8.62i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.603 + 3.42i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 6.03iT - 53T^{2} \) |
| 59 | \( 1 + (1.58 + 0.577i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.69 - 9.63i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.27 + 6.29i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.91 - 11.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.999 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.46 + 2.94i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (9.80 + 8.22i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-8.51 - 4.91i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.2 + 4.83i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.30808811689413977758464031913, −13.05812914458177317974396751180, −11.48449475482558397235228852255, −10.65291596837194515816687033466, −8.944648088389769216046005714596, −8.337244892239179342424004708749, −7.20838510864125168403998078345, −6.39121013178467899721541513943, −4.47874440159217168142823597383, −2.07259630512153067036660041145,
1.97024379420362676984520328253, 4.05888407873372599774081020492, 4.80095921792845225382174267071, 7.68370743161455777587150775780, 8.419008560887662988799963953037, 9.224947953942779691460924570724, 10.46867997856132589282707585012, 11.33017315982231539167479938975, 12.40658968521400944322925328658, 13.55891393025311082923870002813
