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
−13.18027945168394912625400960967, −11.82049188130356442524674077389, −10.48193579654684014039740179792, −9.975738278232458799330084027021, −8.599195323880127248548280507797, −7.84604440991890739053254594377, −6.69806958057561133189156614833, −4.81422590930139713854073495025, −3.87000755462529653979581323571, −2.13815474993861405299726239242,
2.09757629818352275877504583079, 3.41259078870631410097943477117, 5.16340541342841355067030357509, 6.47141940835705618985266775715, 7.954167103250830845687834783583, 8.430942408257370302185555027421, 9.623525153371136573190120735837, 10.67852423937041252165465614445, 12.08291851424609480725974157228, 12.71578425065721881811437572988