L(s) = 1 | + (1.62 − 0.601i)3-s + (0.0726 + 0.125i)5-s + (1.05 + 2.42i)7-s + (2.27 − 1.95i)9-s + (−2.13 − 1.23i)11-s − 2.04i·13-s + (0.193 + 0.160i)15-s + (0.878 − 1.52i)17-s + (−3.68 + 2.12i)19-s + (3.17 + 3.30i)21-s + (−7.46 + 4.30i)23-s + (2.48 − 4.31i)25-s + (2.52 − 4.54i)27-s + 7.08i·29-s + (3.11 + 1.80i)31-s + ⋯ |
L(s) = 1 | + (0.937 − 0.347i)3-s + (0.0324 + 0.0562i)5-s + (0.398 + 0.917i)7-s + (0.758 − 0.651i)9-s + (−0.644 − 0.372i)11-s − 0.566i·13-s + (0.0500 + 0.0414i)15-s + (0.213 − 0.369i)17-s + (−0.844 + 0.487i)19-s + (0.692 + 0.721i)21-s + (−1.55 + 0.898i)23-s + (0.497 − 0.862i)25-s + (0.485 − 0.874i)27-s + 1.31i·29-s + (0.560 + 0.323i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49845 - 0.116921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49845 - 0.116921i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.62 + 0.601i)T \) |
| 7 | \( 1 + (-1.05 - 2.42i)T \) |
good | 5 | \( 1 + (-0.0726 - 0.125i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.13 + 1.23i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.04iT - 13T^{2} \) |
| 17 | \( 1 + (-0.878 + 1.52i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.68 - 2.12i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.46 - 4.30i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.08iT - 29T^{2} \) |
| 31 | \( 1 + (-3.11 - 1.80i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.93 + 5.08i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.33T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 + (-4.65 - 8.05i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.49 + 2.59i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.60 + 9.70i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.66 + 2.69i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.57 + 4.45i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.79iT - 71T^{2} \) |
| 73 | \( 1 + (-11.3 - 6.52i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.86 - 4.95i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + (-4.34 - 7.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.65iT - 97T^{2} \) |
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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
−12.71578425065721881811437572988, −12.08291851424609480725974157228, −10.67852423937041252165465614445, −9.623525153371136573190120735837, −8.430942408257370302185555027421, −7.954167103250830845687834783583, −6.47141940835705618985266775715, −5.16340541342841355067030357509, −3.41259078870631410097943477117, −2.09757629818352275877504583079,
2.13815474993861405299726239242, 3.87000755462529653979581323571, 4.81422590930139713854073495025, 6.69806958057561133189156614833, 7.84604440991890739053254594377, 8.599195323880127248548280507797, 9.975738278232458799330084027021, 10.48193579654684014039740179792, 11.82049188130356442524674077389, 13.18027945168394912625400960967