L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + (−2 − 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s − 7·13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.408·6-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.150 + 0.261i)11-s + (0.144 − 0.249i)12-s − 1.94·13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (0.117 + 0.204i)18-s + (−0.114 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4354050197\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4354050197\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 7T + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−9.559796413988722056828933961406, −9.173266552524022664719450064894, −7.65515762491952398171955998303, −7.09948093994691587905881795124, −5.87742593292052611748775276339, −4.80688969827155091116748630359, −4.15593465151363062779028594369, −3.07752250320498633143776105304, −2.17770690470256555811294470287, −0.15145845653184741597426219134,
2.14682358624930695897044834236, 3.09701589158563214147548231483, 4.25137419982369498222301125032, 5.41795298118006343070344291396, 6.09874473276908627826360804635, 7.11753289541105062079139813968, 7.54655860121012267432293339449, 8.763429218826338024815924245001, 9.217847943863417654910131735616, 10.14177679857201076399853575460