| L(s) = 1 | + (−1.11 − 0.866i)2-s + (1.27 − 1.16i)3-s + (0.496 + 1.93i)4-s + (−2.95 − 1.57i)5-s + (−2.44 + 0.194i)6-s + (−1.29 − 0.257i)7-s + (1.12 − 2.59i)8-s + (0.275 − 2.98i)9-s + (1.93 + 4.32i)10-s + (−0.793 + 0.651i)11-s + (2.89 + 1.89i)12-s + (−0.787 + 0.421i)13-s + (1.22 + 1.40i)14-s + (−5.62 + 1.42i)15-s + (−3.50 + 1.92i)16-s + (−2.18 + 0.903i)17-s + ⋯ |
| L(s) = 1 | + (−0.790 − 0.613i)2-s + (0.738 − 0.673i)3-s + (0.248 + 0.968i)4-s + (−1.32 − 0.705i)5-s + (−0.996 + 0.0794i)6-s + (−0.488 − 0.0972i)7-s + (0.397 − 0.917i)8-s + (0.0918 − 0.995i)9-s + (0.610 + 1.36i)10-s + (−0.239 + 0.196i)11-s + (0.836 + 0.548i)12-s + (−0.218 + 0.116i)13-s + (0.326 + 0.376i)14-s + (−1.45 + 0.368i)15-s + (−0.876 + 0.481i)16-s + (−0.528 + 0.219i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0991446 + 0.394276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0991446 + 0.394276i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.11 + 0.866i)T \) |
| 3 | \( 1 + (-1.27 + 1.16i)T \) |
| good | 5 | \( 1 + (2.95 + 1.57i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (1.29 + 0.257i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.793 - 0.651i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.787 - 0.421i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (2.18 - 0.903i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.960 - 3.16i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (3.39 + 2.26i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (5.01 + 4.11i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (0.756 - 0.756i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.944 + 3.11i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-9.51 - 6.35i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-1.19 + 12.0i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-0.478 + 0.198i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (9.92 - 8.14i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (5.59 + 2.99i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.540 + 5.48i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-11.0 + 1.08i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-13.6 - 2.70i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (0.617 + 3.10i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.69 + 8.92i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-5.26 + 17.3i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-6.31 - 9.45i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (2.77 + 2.77i)T + 97iT^{2} \) |
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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
−10.87333131839571639671789565730, −9.656488475205803289273039878510, −8.913220631657672055879529968679, −7.975931256390183255321900025960, −7.61533275146738922580450390679, −6.42658653233524662008188553053, −4.27947707945570496930483413612, −3.50037492488381031350295586252, −2.05329947946691591581991846718, −0.30428757714159809100683543231,
2.59268024622982106855409804397, 3.78954121443023377434697173718, 5.02214121373265288284238547193, 6.50233551163021965487976339572, 7.51820885555581906953639836848, 8.051040569229427294541952655328, 9.104177087085379319956004007752, 9.805663301353251441952036973717, 10.98828115640226567336281926576, 11.22291983330837442662726821375
