L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.54 − 0.893i)5-s + (0.165 − 2.64i)7-s − 0.999i·8-s + (0.893 + 1.54i)10-s + (−0.692 − 3.24i)11-s + 6.37·13-s + (−1.46 + 2.20i)14-s + (−0.5 + 0.866i)16-s + (−0.0530 − 0.0918i)17-s + (−2.07 + 3.59i)19-s − 1.78i·20-s + (−1.02 + 3.15i)22-s + (3.97 − 6.89i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.692 − 0.399i)5-s + (0.0625 − 0.998i)7-s − 0.353i·8-s + (0.282 + 0.489i)10-s + (−0.208 − 0.977i)11-s + 1.76·13-s + (−0.391 + 0.589i)14-s + (−0.125 + 0.216i)16-s + (−0.0128 − 0.0222i)17-s + (−0.476 + 0.824i)19-s − 0.399i·20-s + (−0.217 + 0.672i)22-s + (0.829 − 1.43i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8397499491\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8397499491\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.165 + 2.64i)T \) |
| 11 | \( 1 + (0.692 + 3.24i)T \) |
good | 5 | \( 1 + (1.54 + 0.893i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 6.37T + 13T^{2} \) |
| 17 | \( 1 + (0.0530 + 0.0918i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.07 - 3.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.97 + 6.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.65iT - 29T^{2} \) |
| 31 | \( 1 + (-1.10 + 0.639i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.26 - 3.91i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.0321T + 41T^{2} \) |
| 43 | \( 1 + 6.87iT - 43T^{2} \) |
| 47 | \( 1 + (8.55 + 4.94i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.313 + 0.543i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (10.7 - 6.22i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.97 + 8.61i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.43 + 9.41i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + (-0.0625 - 0.108i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.80 + 5.08i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + (-4.64 - 2.68i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.7iT - 97T^{2} \) |
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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
−9.003204397639712902070972351458, −8.391884321584211972883959972050, −8.023265795365982552937108151220, −6.85175078681322259194195658095, −6.17887501820128982240953378855, −4.82382125114625442291740263171, −3.81244548027683650743288964218, −3.25237312233066870793055919728, −1.48327248129947942745744645119, −0.45827495872979752731855860689,
1.50303903812243783277212691971, 2.75502250167082419572159348688, 3.86760719858370358055445439412, 5.01797817968309955806307443485, 5.97758174840344260356457282021, 6.70473309301509927040929030535, 7.64146691690003020225274716491, 8.228001258634065854763265871826, 9.117310104343769952383391523100, 9.620356202407366442370463258653