L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.61 + 2.80i)3-s + (−0.499 − 0.866i)4-s + (−1.61 + 2.80i)5-s − 3.23·6-s + 0.999·8-s + (−3.73 + 6.47i)9-s + (−1.61 − 2.80i)10-s + (−0.5 − 0.866i)11-s + (1.61 − 2.80i)12-s + 1.23·13-s − 10.4·15-s + (−0.5 + 0.866i)16-s + (3.23 + 5.60i)17-s + (−3.73 − 6.47i)18-s + (1.38 − 2.39i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.934 + 1.61i)3-s + (−0.249 − 0.433i)4-s + (−0.723 + 1.25i)5-s − 1.32·6-s + 0.353·8-s + (−1.24 + 2.15i)9-s + (−0.511 − 0.886i)10-s + (−0.150 − 0.261i)11-s + (0.467 − 0.809i)12-s + 0.342·13-s − 2.70·15-s + (−0.125 + 0.216i)16-s + (0.784 + 1.35i)17-s + (−0.880 − 1.52i)18-s + (0.317 − 0.549i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.320687618\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320687618\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-1.61 - 2.80i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.61 - 2.80i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + (-3.23 - 5.60i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.38 + 2.39i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.47 + 9.47i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.236 - 0.408i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.61 + 6.26i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.61 + 4.53i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.70 - 13.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + (-2.47 - 4.28i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−10.25112248338798068207408983133, −9.579410063564343782191275243783, −8.801855244583854520503490494316, −7.900450737225095132405297369434, −7.51148348895111774359892507744, −6.13561696880837170244503892386, −5.27911606995012011257866548434, −3.87936246501667235564082525803, −3.67350723487970640813544777321, −2.45247768376633488459939643326,
0.62274468191681132681374089533, 1.47440520033644363541171751711, 2.68056070979100287332154591508, 3.64336587891415512283406583339, 4.83929534306983776112955265874, 6.11971204866869489111216787781, 7.36816339226097863728342080878, 7.78628572518744370547544682960, 8.457789018410234657318521314520, 9.109875962892410110040575523730