| L(s) = 1 | + (0.692 + 1.23i)2-s + (−1.18 + 2.59i)3-s + (−1.04 + 1.70i)4-s + (−1.01 + 1.17i)5-s + (−4.02 + 0.335i)6-s + (4.25 − 2.73i)7-s + (−2.82 − 0.101i)8-s + (−3.37 − 3.89i)9-s + (−2.14 − 0.441i)10-s + (1.77 + 0.254i)11-s + (−3.20 − 4.73i)12-s + (−0.511 + 0.796i)13-s + (6.31 + 3.35i)14-s + (−1.84 − 4.03i)15-s + (−1.83 − 3.55i)16-s + (−1.67 + 5.70i)17-s + ⋯ |
| L(s) = 1 | + (0.489 + 0.871i)2-s + (−0.685 + 1.50i)3-s + (−0.520 + 0.853i)4-s + (−0.454 + 0.524i)5-s + (−1.64 + 0.137i)6-s + (1.60 − 1.03i)7-s + (−0.999 − 0.0358i)8-s + (−1.12 − 1.29i)9-s + (−0.679 − 0.139i)10-s + (0.534 + 0.0768i)11-s + (−0.924 − 1.36i)12-s + (−0.141 + 0.220i)13-s + (1.68 + 0.896i)14-s + (−0.475 − 1.04i)15-s + (−0.457 − 0.888i)16-s + (−0.406 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.104090 + 1.15317i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.104090 + 1.15317i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.692 - 1.23i)T \) |
| 23 | \( 1 + (-3.08 - 3.67i)T \) |
| good | 3 | \( 1 + (1.18 - 2.59i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (1.01 - 1.17i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-4.25 + 2.73i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-1.77 - 0.254i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.511 - 0.796i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.67 - 5.70i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.812 + 2.76i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.87 + 6.39i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.15 + 0.982i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.98 + 4.59i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (3.66 - 4.23i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-5.98 - 2.73i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 4.26iT - 47T^{2} \) |
| 53 | \( 1 + (-3.03 + 1.94i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.53 - 1.63i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.27 - 4.99i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 1.50i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (2.06 - 0.296i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.05 - 0.603i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (6.55 + 4.21i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 5.25i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.48 - 1.13i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.549 - 0.476i)T + (13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.37474599372193951883008912112, −11.69213401135907874846042374811, −11.21616919026086462055761646346, −10.31552952761149709727907537715, −8.980747825782947951292187472582, −7.87049213931517721055818952575, −6.75769500828433282112492818607, −5.37223811187821526009440401016, −4.37346824196515804514713576596, −3.84461015300003628172809360827,
1.12404052924364586921225955378, 2.39051661307952002913044823853, 4.73992155456041767119333054294, 5.45860975469214854996111424253, 6.78582585480521288365568927196, 8.185445662315451281705439486887, 8.910882439271398607000568008093, 10.79784251720669952972623676239, 11.64303605211518167504521965504, 12.10617433170296128854638515912
