| L(s) = 1 | + (−0.688 − 1.23i)2-s + (1.98 − 0.999i)3-s + (−1.05 + 1.70i)4-s + (−1.44 − 3.26i)5-s + (−2.60 − 1.76i)6-s + (2.80 + 1.67i)7-s + (2.82 + 0.129i)8-s + (1.15 − 1.55i)9-s + (−3.03 + 4.02i)10-s + (−2.51 − 1.96i)11-s + (−0.389 + 4.42i)12-s + (3.77 − 3.59i)13-s + (0.146 − 4.61i)14-s + (−6.12 − 5.02i)15-s + (−1.78 − 3.57i)16-s + (1.61 + 1.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.486 − 0.873i)2-s + (1.14 − 0.576i)3-s + (−0.526 + 0.850i)4-s + (−0.646 − 1.45i)5-s + (−1.06 − 0.720i)6-s + (1.05 + 0.634i)7-s + (0.998 + 0.0458i)8-s + (0.384 − 0.518i)9-s + (−0.959 + 1.27i)10-s + (−0.759 − 0.592i)11-s + (−0.112 + 1.27i)12-s + (1.04 − 0.997i)13-s + (0.0390 − 1.23i)14-s + (−1.58 − 1.29i)15-s + (−0.446 − 0.894i)16-s + (0.391 + 0.477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.458238 - 1.39314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.458238 - 1.39314i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.688 + 1.23i)T \) |
| good | 3 | \( 1 + (-1.98 + 0.999i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (1.44 + 3.26i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-2.80 - 1.67i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (2.51 + 1.96i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.77 + 3.59i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.61 - 1.96i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.451 + 2.60i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (5.78 + 2.73i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (1.85 + 1.05i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (1.82 - 0.363i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-7.69 - 4.87i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (4.03 + 4.45i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-3.67 - 11.1i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-3.68 - 1.97i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-7.73 + 4.38i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (4.42 - 4.64i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.692 + 9.39i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (4.47 + 3.86i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-16.1 - 2.39i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (8.02 - 4.80i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (0.635 + 0.192i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-6.18 + 3.91i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-1.04 + 0.493i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 6.77i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.65854995843093255691792267703, −9.304429898935104441079988010852, −8.514018229019233234285317662788, −8.094444288604915122555930185668, −7.87609608217032648310849155187, −5.60669119183236976800246969160, −4.53649768531600436298710989303, −3.37507809678585813906721642911, −2.16485486056055799634914259270, −0.975058023563848701559090176063,
2.08268209005210706839218546832, 3.70415918723132506725598919693, 4.32702386481630275901880739713, 5.87570652696419980073963789133, 7.16842180935330458385730795870, 7.66910173879684403217299764221, 8.335325773946975025330333157266, 9.375830635440527515781497697483, 10.28459966700567825182350411238, 10.84142953690854338984345497985
