| L(s) = 1 | + (1.36 + 0.382i)2-s + (−1.09 − 2.39i)3-s + (1.70 + 1.04i)4-s + (−2.07 − 2.39i)5-s + (−0.574 − 3.67i)6-s + (−2.05 − 1.32i)7-s + (1.92 + 2.06i)8-s + (−2.57 + 2.97i)9-s + (−1.91 − 4.06i)10-s + (4.22 − 0.606i)11-s + (0.623 − 5.23i)12-s + (1.73 + 2.69i)13-s + (−2.29 − 2.58i)14-s + (−3.47 + 7.60i)15-s + (1.83 + 3.55i)16-s + (1.19 + 4.05i)17-s + ⋯ |
| L(s) = 1 | + (0.962 + 0.270i)2-s + (−0.631 − 1.38i)3-s + (0.854 + 0.520i)4-s + (−0.929 − 1.07i)5-s + (−0.234 − 1.50i)6-s + (−0.777 − 0.499i)7-s + (0.681 + 0.731i)8-s + (−0.859 + 0.991i)9-s + (−0.605 − 1.28i)10-s + (1.27 − 0.182i)11-s + (0.180 − 1.50i)12-s + (0.480 + 0.747i)13-s + (−0.613 − 0.690i)14-s + (−0.896 + 1.96i)15-s + (0.458 + 0.888i)16-s + (0.288 + 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09257 - 0.958599i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09257 - 0.958599i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.382i)T \) |
| 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
−12.62444872957842441299273030837, −11.67937388856997921464403635435, −11.23020584733690267410262948662, −9.060614288384482371212051823047, −7.912915139415462253079921094520, −6.84024661847004290877340969868, −6.33879196717214191727332058631, −4.81061025836523642152272359163, −3.61844873674530655634072589782, −1.23001545978486178299538724402,
3.35936867987940159198490248179, 3.70914303759870345367742928749, 5.21876176044477982797953744304, 6.23068626284826175630730658938, 7.31772652096895368821901631308, 9.299947357774016947458621699166, 10.23855515069446040518825426601, 11.00855432088610057089326629897, 11.75626655666011026925593217347, 12.47659573297113631756276365491
