| L(s) = 1 | + (−1.41 − 0.0170i)2-s + (−1.09 − 2.39i)3-s + (1.99 + 0.0482i)4-s + (2.07 + 2.39i)5-s + (1.50 + 3.40i)6-s + (2.05 + 1.32i)7-s + (−2.82 − 0.102i)8-s + (−2.57 + 2.97i)9-s + (−2.89 − 3.42i)10-s + (4.22 − 0.606i)11-s + (−2.07 − 4.84i)12-s + (−1.73 − 2.69i)13-s + (−2.88 − 1.90i)14-s + (3.47 − 7.60i)15-s + (3.99 + 0.192i)16-s + (1.19 + 4.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0120i)2-s + (−0.631 − 1.38i)3-s + (0.999 + 0.0241i)4-s + (0.929 + 1.07i)5-s + (0.614 + 1.39i)6-s + (0.777 + 0.499i)7-s + (−0.999 − 0.0361i)8-s + (−0.859 + 0.991i)9-s + (−0.916 − 1.08i)10-s + (1.27 − 0.182i)11-s + (−0.598 − 1.39i)12-s + (−0.480 − 0.747i)13-s + (−0.771 − 0.508i)14-s + (0.896 − 1.96i)15-s + (0.998 + 0.0482i)16-s + (0.288 + 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.768287 - 0.261392i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.768287 - 0.261392i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0170i)T \) |
| 23 | \( 1 + (4.76 - 0.516i)T \) |
| good | 3 | \( 1 + (1.09 + 2.39i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-2.07 - 2.39i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-2.05 - 1.32i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-4.22 + 0.606i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.73 + 2.69i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.19 - 4.05i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.01 + 6.85i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.219 - 0.746i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.02 - 1.38i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.72 + 4.29i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.46 - 5.14i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (7.42 - 3.39i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 4.96iT - 47T^{2} \) |
| 53 | \( 1 + (-3.91 - 2.51i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.13 - 5.22i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.48 - 5.43i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.79 - 0.401i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (8.76 + 1.26i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.107 - 0.0317i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (3.79 - 2.43i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.77 - 4.13i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.30 + 1.51i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.10 + 2.69i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.15157001892022178372733948557, −11.52310387081134768782280307144, −10.66351807715908941785455358305, −9.554140938412697212635970979639, −8.288602152676549359177631621660, −7.25803321823360945692897013527, −6.43247608085455429485324265850, −5.74005246644548248610047300372, −2.62347858516444427797682990843, −1.45557495326731200020089722169,
1.51605166966720598162871486083, 4.11559362333785046927011883328, 5.20778112985482284948441877637, 6.29369734419640430199159366137, 7.901386076805318097855211956337, 9.228993680993453881376693134399, 9.626582796232629538984282587079, 10.39413983777448903105065703999, 11.65567898778440216983376834383, 12.07474266926843383186236687532
