| L(s) = 1 | + (−0.544 + 1.57i)2-s + (−0.458 − 0.888i)3-s + (−0.604 − 0.475i)4-s + (1.92 − 0.183i)5-s + (1.64 − 0.236i)6-s + (0.763 + 2.53i)7-s + (−1.72 + 1.10i)8-s + (−0.580 + 0.814i)9-s + (−0.758 + 3.12i)10-s + (3.76 − 1.30i)11-s + (−0.145 + 0.755i)12-s + (−0.663 − 2.25i)13-s + (−4.39 − 0.177i)14-s + (−1.04 − 1.62i)15-s + (−1.16 − 4.80i)16-s + (3.90 + 1.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.384 + 1.11i)2-s + (−0.264 − 0.513i)3-s + (−0.302 − 0.237i)4-s + (0.860 − 0.0821i)5-s + (0.672 − 0.0966i)6-s + (0.288 + 0.957i)7-s + (−0.609 + 0.391i)8-s + (−0.193 + 0.271i)9-s + (−0.239 + 0.988i)10-s + (1.13 − 0.393i)11-s + (−0.0420 + 0.218i)12-s + (−0.183 − 0.626i)13-s + (−1.17 − 0.0473i)14-s + (−0.269 − 0.419i)15-s + (−0.291 − 1.20i)16-s + (0.947 + 0.379i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.863291 + 1.00169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.863291 + 1.00169i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.458 + 0.888i)T \) |
| 7 | \( 1 + (-0.763 - 2.53i)T \) |
| 23 | \( 1 + (-1.51 - 4.54i)T \) |
| good | 2 | \( 1 + (0.544 - 1.57i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (-1.92 + 0.183i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-3.76 + 1.30i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (0.663 + 2.25i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-3.90 - 1.56i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (1.75 - 0.700i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-1.49 - 10.4i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.19 + 0.342i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (8.71 + 6.20i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-2.91 - 1.32i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.00 + 1.56i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (5.05 - 2.92i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.03 + 2.13i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-10.2 - 2.48i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-3.70 - 1.91i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (1.43 + 7.42i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (4.63 + 5.34i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (4.96 - 6.30i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-1.71 + 1.79i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (0.835 + 1.82i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.785 + 16.4i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (4.00 - 8.76i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.37385102761621434358266045783, −10.09817073701206828190845220867, −9.058025768955185160536904680740, −8.513587989067779997176063856301, −7.52188518827084115469676800710, −6.54332111761039629041444737091, −5.80252082831845957211464771837, −5.26054523408107539454790359941, −3.13100314513216699195766279053, −1.62297120033587154156479772844,
1.06441553298326040871682388906, 2.33608565412012569443774382329, 3.76240634238162522634751974381, 4.67709584188912875597704374387, 6.18480598961582258272865955169, 6.84727589139432492916906332932, 8.401739213376689939531350725140, 9.520834054134581675688738547878, 9.965450199182374468717671567225, 10.52207580818438352269142783415
