| L(s) = 1 | + (−2.57 + 1.01i)2-s + (1.30 + 1.13i)3-s + (4.14 − 3.84i)4-s + (3.17 + 1.52i)5-s + (−4.51 − 1.60i)6-s + (2.46 − 0.972i)7-s + (−4.39 + 9.11i)8-s + (0.414 + 2.97i)9-s + (−9.72 − 0.728i)10-s + (−0.673 − 0.537i)11-s + (9.79 − 0.312i)12-s + (−3.72 + 1.46i)13-s + (−5.35 + 4.99i)14-s + (2.41 + 5.60i)15-s + (1.24 − 16.6i)16-s + (1.51 + 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−1.82 + 0.714i)2-s + (0.754 + 0.656i)3-s + (2.07 − 1.92i)4-s + (1.42 + 0.683i)5-s + (−1.84 − 0.656i)6-s + (0.930 − 0.367i)7-s + (−1.55 + 3.22i)8-s + (0.138 + 0.990i)9-s + (−3.07 − 0.230i)10-s + (−0.203 − 0.161i)11-s + (2.82 − 0.0902i)12-s + (−1.03 + 0.405i)13-s + (−1.43 + 1.33i)14-s + (0.622 + 1.44i)15-s + (0.311 − 4.16i)16-s + (0.368 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.713257 + 0.800173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.713257 + 0.800173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.30 - 1.13i)T \) |
| 7 | \( 1 + (-2.46 + 0.972i)T \) |
| good | 2 | \( 1 + (2.57 - 1.01i)T + (1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-3.17 - 1.52i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (0.673 + 0.537i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (3.72 - 1.46i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-1.51 - 1.40i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.72 + 0.995i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.25 + 0.286i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (2.61 + 8.48i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-2.40 + 1.38i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.25 + 0.387i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (4.21 - 2.87i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (3.57 + 2.43i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (1.71 + 4.37i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (3.02 - 9.81i)T + (-43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (3.20 + 2.18i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (-2.21 + 2.38i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.29 - 5.71i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.0685 + 0.0156i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (0.591 + 3.92i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (4.44 - 7.70i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.43 + 11.2i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-0.985 + 2.50i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (0.845 - 0.488i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.76963200766509408689934337830, −9.947841213929711891966103302340, −9.741520670815735996891116616634, −8.732435859413872476007641124964, −7.84025471893235556111119656601, −7.13429996202459919077645902176, −5.99143696887531765025369027341, −5.00565930700127872516899331479, −2.60293386515701102411522514570, −1.77183801695687547695799694010,
1.26267308260467253609640384285, 2.03753299691541289875415569715, 2.99857712646512205307751097746, 5.27112246348599377193109619362, 6.71900363514154983059906462526, 7.66447299374437127976904474437, 8.373197209225577514147075731735, 9.186515360811214836918543173583, 9.683342833697712357025425224062, 10.50812351005973330783324267988
