| 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 | \( 1 + (-0.549 + 0.476i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.10617433170296128854638515912, −11.64303605211518167504521965504, −10.79784251720669952972623676239, −8.910882439271398607000568008093, −8.185445662315451281705439486887, −6.78582585480521288365568927196, −5.45860975469214854996111424253, −4.73992155456041767119333054294, −2.39051661307952002913044823853, −1.12404052924364586921225955378,
3.84461015300003628172809360827, 4.37346824196515804514713576596, 5.37223811187821526009440401016, 6.75769500828433282112492818607, 7.87049213931517721055818952575, 8.980747825782947951292187472582, 10.31552952761149709727907537715, 11.21616919026086462055761646346, 11.69213401135907874846042374811, 13.37474599372193951883008912112
