L(s) = 1 | + (−0.707 − 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (−3.62 + 2.09i)5-s − 2.44i·6-s + (1.62 + 2.09i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (5.12 + 2.95i)10-s + (0.0857 − 0.148i)11-s + (−2.99 + 1.73i)12-s + (1.41 − 3.46i)14-s − 7.24·15-s + (−2.00 − 3.46i)16-s + (2.12 − 3.67i)18-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)2-s + (0.866 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (−1.61 + 0.935i)5-s − 0.999i·6-s + (0.612 + 0.790i)7-s + 0.999·8-s + (0.5 + 0.866i)9-s + (1.61 + 0.935i)10-s + (0.0258 − 0.0448i)11-s + (−0.866 + 0.500i)12-s + (0.377 − 0.925i)14-s − 1.87·15-s + (−0.500 − 0.866i)16-s + (0.499 − 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.834312 + 0.346049i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.834312 + 0.346049i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.62 - 2.09i)T \) |
good | 5 | \( 1 + (3.62 - 2.09i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.0857 + 0.148i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + (5.37 + 3.10i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.03 + 8.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.98 - 2.30i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-8.48 - 4.89i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.86 + 6.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.76iT - 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−12.58519226030291841966644791953, −11.66319167707082652482350229044, −10.99289635652748039291487779932, −10.03970845039765290764869198924, −8.690658367883543879252176892806, −8.137653966365733706642113049803, −7.19625598872623055651045968226, −4.64103339255794857769084713322, −3.58602435242247003626243127541, −2.53490431193428624634432690307,
1.02144656163216996523133460833, 3.89308266053694064024751807343, 4.85394360141271860217538016938, 6.87418260018588715958806117078, 7.70343024205803142836580164284, 8.306887473294608186943884842203, 9.081614607006918424209978112388, 10.52625358378780646427886621938, 11.79681366978133850406297616697, 12.80849472640676188843890857589