L(s) = 1 | + (−1.65 − 0.524i)3-s + (−1.57 + 2.72i)5-s + (2.21 − 1.27i)7-s + (2.44 + 1.73i)9-s + (−2.02 + 1.16i)11-s + (−2.59 − 1.5i)13-s + (4.02 − 3.67i)15-s + 4.24i·17-s + 8.08·19-s + (−4.33 + 0.949i)21-s + (−0.642 + 1.11i)23-s + (−2.44 − 4.24i)25-s + (−3.13 − 4.14i)27-s + (−1.18 − 2.05i)29-s + (−7.64 − 4.41i)31-s + ⋯ |
L(s) = 1 | + (−0.953 − 0.302i)3-s + (−0.703 + 1.21i)5-s + (0.837 − 0.483i)7-s + (0.816 + 0.577i)9-s + (−0.609 + 0.351i)11-s + (−0.720 − 0.416i)13-s + (1.03 − 0.948i)15-s + 1.02i·17-s + 1.85·19-s + (−0.944 + 0.207i)21-s + (−0.133 + 0.232i)23-s + (−0.489 − 0.848i)25-s + (−0.603 − 0.797i)27-s + (−0.219 − 0.380i)29-s + (−1.37 − 0.792i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4168259747\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4168259747\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 + 0.524i)T \) |
good | 5 | \( 1 + (1.57 - 2.72i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.21 + 1.27i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.02 - 1.16i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.59 + 1.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 8.08T + 19T^{2} \) |
| 23 | \( 1 + (0.642 - 1.11i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.18 + 2.05i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.64 + 4.41i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.34iT - 37T^{2} \) |
| 41 | \( 1 + (8.17 + 4.71i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.78 - 8.29i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.34T + 53T^{2} \) |
| 59 | \( 1 + (-1.11 - 0.642i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.79 - 4.5i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.204 - 0.353i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 1.55T + 73T^{2} \) |
| 79 | \( 1 + (8.93 - 5.15i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.93 - 1.69i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.14iT - 89T^{2} \) |
| 97 | \( 1 + (-6.62 - 11.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.42511655986475248760669252569, −9.673567050423306893517749646897, −8.013262173906008916824384364286, −7.50877273094942853039564758845, −7.10670569246425683505769799671, −5.91514427927154738684474553126, −5.11171909554399710290585493059, −4.13417265045190705624909007498, −3.00217102780406206961024173823, −1.57708080825743907592486580265,
0.21445120519774850687567889132, 1.55406616902725920494765407737, 3.33020538114138564339480918022, 4.64149445345605793124175583163, 5.08208034635013508012938201193, 5.61662643480834511260070654906, 7.15429630346215731826783302602, 7.68827090004073167672071628869, 8.776999226327372321228413645281, 9.342853916859805522309319828502