| L(s) = 1 | + (0.580 − 1.28i)2-s + (−1.63 + 0.575i)3-s + (−1.32 − 1.49i)4-s + (3.21 + 1.17i)5-s + (−0.206 + 2.44i)6-s + (2.85 + 0.504i)7-s + (−2.70 + 0.841i)8-s + (2.33 − 1.87i)9-s + (3.37 − 3.46i)10-s + (−0.927 − 2.54i)11-s + (3.02 + 1.68i)12-s + (−1.60 − 1.91i)13-s + (2.30 − 3.39i)14-s + (−5.92 − 0.0624i)15-s + (−0.483 + 3.97i)16-s + (4.30 − 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.410 − 0.911i)2-s + (−0.943 + 0.332i)3-s + (−0.663 − 0.748i)4-s + (1.43 + 0.523i)5-s + (−0.0843 + 0.996i)6-s + (1.08 + 0.190i)7-s + (−0.954 + 0.297i)8-s + (0.779 − 0.626i)9-s + (1.06 − 1.09i)10-s + (−0.279 − 0.768i)11-s + (0.874 + 0.485i)12-s + (−0.444 − 0.530i)13-s + (0.617 − 0.907i)14-s + (−1.53 − 0.0161i)15-s + (−0.120 + 0.992i)16-s + (1.04 − 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20593 - 0.666357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20593 - 0.666357i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.580 + 1.28i)T \) |
| 3 | \( 1 + (1.63 - 0.575i)T \) |
| good | 5 | \( 1 + (-3.21 - 1.17i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-2.85 - 0.504i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.927 + 2.54i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.60 + 1.91i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.30 + 2.48i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.31 - 2.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.157 + 0.895i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.00 - 5.87i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.90 - 0.512i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (9.78 - 5.64i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.04 + 3.62i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.12 - 2.23i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.225 + 1.28i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + (-2.96 + 8.13i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.720 + 0.127i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.00 + 4.19i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.18 - 8.97i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.34 - 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.64 - 6.73i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.15 + 3.75i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (4.90 + 2.82i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.126 - 0.0461i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.04445080891047054560513861284, −11.11743613098486381498352166010, −10.33139225566149229027869402228, −9.856224321788049806177471187162, −8.482242218239742441644492800499, −6.59823202780269115591497374176, −5.42210929561615842954152738507, −5.09030962213909467881168398618, −3.15848237924391051783973978022, −1.54842774334141802511149863770,
1.81155771823139914855060780271, 4.60100246688617356207923005037, 5.20905863625862511325377993851, 6.14956828337665333428224514784, 7.19896311835096514641920753508, 8.246178028097893794165370675752, 9.550444774669931179639464146190, 10.43252516236878614517183987253, 11.88102188013820075685303519718, 12.58225474991458947684982322867
