| L(s) = 1 | + (−0.321 − 1.82i)2-s + (−2.27 + 1.59i)3-s + (−1.33 + 0.485i)4-s + (1.54 + 1.62i)5-s + (3.63 + 3.63i)6-s + (−0.309 − 3.54i)7-s + (−0.535 − 0.927i)8-s + (1.61 − 4.44i)9-s + (2.45 − 3.32i)10-s + (4.83 − 2.79i)11-s + (2.26 − 3.23i)12-s + (2.98 − 1.08i)13-s + (−6.35 + 1.70i)14-s + (−6.09 − 1.23i)15-s + (−3.69 + 3.10i)16-s + (−1.03 + 2.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.227 − 1.28i)2-s + (−1.31 + 0.920i)3-s + (−0.667 + 0.242i)4-s + (0.688 + 0.724i)5-s + (1.48 + 1.48i)6-s + (−0.117 − 1.33i)7-s + (−0.189 − 0.327i)8-s + (0.539 − 1.48i)9-s + (0.777 − 1.05i)10-s + (1.45 − 0.841i)11-s + (0.654 − 0.934i)12-s + (0.826 − 0.300i)13-s + (−1.69 + 0.454i)14-s + (−1.57 − 0.318i)15-s + (−0.923 + 0.775i)16-s + (−0.250 + 0.687i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00177 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00177 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.571769 - 0.572786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.571769 - 0.572786i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.54 - 1.62i)T \) |
| 37 | \( 1 + (1.74 - 5.82i)T \) |
| good | 2 | \( 1 + (0.321 + 1.82i)T + (-1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (2.27 - 1.59i)T + (1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (0.309 + 3.54i)T + (-6.89 + 1.21i)T^{2} \) |
| 11 | \( 1 + (-4.83 + 2.79i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.98 + 1.08i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.03 - 2.83i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.0922 - 0.131i)T + (-6.49 + 17.8i)T^{2} \) |
| 23 | \( 1 + (-3.22 + 5.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.02 + 7.54i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (3.60 - 3.60i)T - 31iT^{2} \) |
| 41 | \( 1 + (-0.214 - 0.589i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + 0.198T + 43T^{2} \) |
| 47 | \( 1 + (0.516 - 0.138i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.212 - 2.42i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (0.179 - 2.05i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (0.187 - 0.402i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (14.0 - 1.22i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.219 + 1.24i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.87 - 6.87i)T + 73iT^{2} \) |
| 79 | \( 1 + (6.64 - 0.581i)T + (77.7 - 13.7i)T^{2} \) |
| 83 | \( 1 + (-8.81 + 4.10i)T + (53.3 - 63.5i)T^{2} \) |
| 89 | \( 1 + (5.12 + 0.448i)T + (87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (-6.38 - 3.68i)T + (48.5 + 84.0i)T^{2} \) |
| 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
−11.81543058209468086229220310612, −11.07839669458064466161443419881, −10.57518803604581451094215424587, −10.04758328245548324145035911203, −8.942877338220722935780363492706, −6.64022347215988481552181058166, −6.13471312290982089408612191182, −4.27267967288401186051940823479, −3.43729657091494257988568598222, −1.05633573304478188498859260837,
1.71351711192762998444250044554, 5.01557043702595828263570630850, 5.77171251770512031793303852577, 6.48330673676798989946121044431, 7.29207670666694046484095445028, 8.872799083974662158379834436256, 9.292435402211047718690982607122, 11.29329210926375725225485446165, 11.98961012701828528315776165804, 12.70489188772767877705997446017
