L(s) = 1 | + (−0.394 − 1.35i)2-s + (1.75 − 1.01i)3-s + (−1.68 + 1.07i)4-s + (0.5 − 0.866i)5-s + (−2.06 − 1.98i)6-s + (1.63 − 2.07i)7-s + (2.12 + 1.86i)8-s + (0.552 − 0.957i)9-s + (−1.37 − 0.336i)10-s + (−0.572 − 0.991i)11-s + (−1.87 + 3.59i)12-s + 0.714·13-s + (−3.46 − 1.40i)14-s − 2.02i·15-s + (1.69 − 3.62i)16-s + (−1.98 + 1.14i)17-s + ⋯ |
L(s) = 1 | + (−0.279 − 0.960i)2-s + (1.01 − 0.584i)3-s + (−0.843 + 0.536i)4-s + (0.223 − 0.387i)5-s + (−0.844 − 0.809i)6-s + (0.619 − 0.785i)7-s + (0.750 + 0.660i)8-s + (0.184 − 0.319i)9-s + (−0.434 − 0.106i)10-s + (−0.172 − 0.299i)11-s + (−0.541 + 1.03i)12-s + 0.198·13-s + (−0.926 − 0.375i)14-s − 0.523i·15-s + (0.424 − 0.905i)16-s + (−0.480 + 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.777496 - 1.26552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.777496 - 1.26552i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.394 + 1.35i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.63 + 2.07i)T \) |
good | 3 | \( 1 + (-1.75 + 1.01i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.572 + 0.991i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.714T + 13T^{2} \) |
| 17 | \( 1 + (1.98 - 1.14i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.36 + 1.94i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.00 - 2.31i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.54iT - 29T^{2} \) |
| 31 | \( 1 + (-0.590 - 1.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.72 + 3.30i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + (2.60 - 4.51i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.25 - 1.30i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.5 + 6.07i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.13 + 12.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.08 - 7.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.8iT - 71T^{2} \) |
| 73 | \( 1 + (4.71 - 2.72i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.20 - 4.73i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.78iT - 83T^{2} \) |
| 89 | \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.18iT - 97T^{2} \) |
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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
−11.33101790853348325516502425910, −10.78312980034471179546264540299, −9.562087084741015389891947054643, −8.613849064749751482524546392271, −8.092236573816307129207484786523, −7.01410173775255249004956570377, −5.09003366452998820867796988213, −3.89556345847262608271673524353, −2.56506664643122097158223956345, −1.33315747738696495727544556301,
2.34107107464017420347527152053, 3.95986726759840117900774511533, 5.12518538539941992552028359437, 6.28534028133670193891502515239, 7.46038110400273581607806431226, 8.586209154841957978327026135846, 8.930212685239043528788970728058, 9.984883857135290608342867342510, 10.86689896547200209682787158538, 12.30192980837439796317242829222