L(s) = 1 | + (−1.32 + 0.480i)2-s + (−0.316 + 1.70i)3-s + (1.53 − 1.27i)4-s + (−1.86 − 3.22i)5-s + (−0.397 − 2.41i)6-s + (−2.46 + 0.962i)7-s + (−1.43 + 2.44i)8-s + (−2.79 − 1.07i)9-s + (4.02 + 3.39i)10-s + (−2.26 − 1.31i)11-s + (1.69 + 3.02i)12-s − 3.57i·13-s + (2.81 − 2.46i)14-s + (6.07 − 2.14i)15-s + (0.729 − 3.93i)16-s + (0.186 + 0.107i)17-s + ⋯ |
L(s) = 1 | + (−0.940 + 0.339i)2-s + (−0.182 + 0.983i)3-s + (0.768 − 0.639i)4-s + (−0.832 − 1.44i)5-s + (−0.162 − 0.986i)6-s + (−0.931 + 0.363i)7-s + (−0.505 + 0.862i)8-s + (−0.933 − 0.359i)9-s + (1.27 + 1.07i)10-s + (−0.684 − 0.395i)11-s + (0.488 + 0.872i)12-s − 0.990i·13-s + (0.752 − 0.658i)14-s + (1.56 − 0.554i)15-s + (0.182 − 0.983i)16-s + (0.0451 + 0.0260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.105762 - 0.160399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.105762 - 0.160399i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.32 - 0.480i)T \) |
| 3 | \( 1 + (0.316 - 1.70i)T \) |
| 7 | \( 1 + (2.46 - 0.962i)T \) |
good | 5 | \( 1 + (1.86 + 3.22i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.26 + 1.31i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.57iT - 13T^{2} \) |
| 17 | \( 1 + (-0.186 - 0.107i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.14 + 1.97i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.33 - 4.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 + (4.26 + 2.46i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.31 - 4.22i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.909iT - 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + (-0.586 - 1.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.06 + 1.83i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.79 + 3.92i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.301 + 0.173i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.98 - 8.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + (-4.45 + 7.71i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.70 + 5.60i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.73iT - 83T^{2} \) |
| 89 | \( 1 + (-15.4 + 8.93i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−12.27984494971258602961153321764, −11.28340488488244359183917971559, −10.27762038518581624112613722385, −9.233393348858892911742994025963, −8.661243168949474366401592467873, −7.62722864277541408542748901084, −5.84737127565929793058598661622, −5.03270351821405601749247903769, −3.29057538252528440483170931365, −0.23062545579855576336669478542,
2.38671658647423532142383804052, 3.57954664686268914428477652317, 6.34970170638846564113117545854, 7.08687357950759094983457931467, 7.64685046380791109223534596377, 9.002502134901830045434750813770, 10.43919598940872457354071355407, 10.93846038625686623257911134750, 12.02131517779405690476268161391, 12.72539427646272286834934342384