| L(s) = 1 | + (1.11 − 0.106i)2-s + (−0.841 + 0.540i)3-s + (−0.742 + 0.143i)4-s + (−0.436 + 3.03i)5-s + (−0.876 + 0.689i)6-s + (0.656 + 0.921i)7-s + (−2.94 + 0.866i)8-s + (0.415 − 0.909i)9-s + (−0.162 + 3.41i)10-s + (2.03 + 1.59i)11-s + (0.547 − 0.522i)12-s + (0.119 + 0.493i)13-s + (0.826 + 0.953i)14-s + (−1.27 − 2.79i)15-s + (−1.77 + 0.711i)16-s + (−0.655 − 0.126i)17-s + ⋯ |
| L(s) = 1 | + (0.784 − 0.0749i)2-s + (−0.485 + 0.312i)3-s + (−0.371 + 0.0715i)4-s + (−0.195 + 1.35i)5-s + (−0.357 + 0.281i)6-s + (0.248 + 0.348i)7-s + (−1.04 + 0.306i)8-s + (0.138 − 0.303i)9-s + (−0.0515 + 1.08i)10-s + (0.613 + 0.482i)11-s + (0.158 − 0.150i)12-s + (0.0331 + 0.136i)13-s + (0.220 + 0.254i)14-s + (−0.329 − 0.721i)15-s + (−0.444 + 0.177i)16-s + (−0.159 − 0.0306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0119 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0119 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.868257 + 0.857979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.868257 + 0.857979i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.841 - 0.540i)T \) |
| 67 | \( 1 + (-8.17 - 0.468i)T \) |
| good | 2 | \( 1 + (-1.11 + 0.106i)T + (1.96 - 0.378i)T^{2} \) |
| 5 | \( 1 + (0.436 - 3.03i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.656 - 0.921i)T + (-2.28 + 6.61i)T^{2} \) |
| 11 | \( 1 + (-2.03 - 1.59i)T + (2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.119 - 0.493i)T + (-11.5 + 5.95i)T^{2} \) |
| 17 | \( 1 + (0.655 + 0.126i)T + (15.7 + 6.31i)T^{2} \) |
| 19 | \( 1 + (-2.28 + 3.21i)T + (-6.21 - 17.9i)T^{2} \) |
| 23 | \( 1 + (-4.23 - 2.18i)T + (13.3 + 18.7i)T^{2} \) |
| 29 | \( 1 + (-2.53 + 4.38i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.447 - 1.84i)T + (-27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (5.95 + 10.3i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.32 - 6.71i)T + (-32.2 + 25.3i)T^{2} \) |
| 43 | \( 1 + (2.52 - 2.91i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-0.517 - 10.8i)T + (-46.7 + 4.46i)T^{2} \) |
| 53 | \( 1 + (-2.68 - 3.09i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-9.97 + 2.92i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (0.0373 - 0.0293i)T + (14.3 - 59.2i)T^{2} \) |
| 71 | \( 1 + (12.8 - 2.47i)T + (65.9 - 26.3i)T^{2} \) |
| 73 | \( 1 + (0.950 - 0.747i)T + (17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-0.693 + 0.660i)T + (3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-5.90 + 2.36i)T + (60.0 - 57.2i)T^{2} \) |
| 89 | \( 1 + (-1.69 - 1.08i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (7.91 + 13.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.63861568386140759062742168742, −11.64285105421010303664120905802, −11.10410480791962961358234248646, −9.841742659683466947201968347385, −8.894841680202304850374352108699, −7.30117226086792805418603438268, −6.33070368522589438543273443028, −5.16036159283608521338732787350, −4.02071396561563831651287556782, −2.84131991153014095491576260211,
0.987501573099413778018044099543, 3.70939508689427725270508136185, 4.83116199126963015299795650789, 5.52733030299384823864950771826, 6.81201469989724060977738500313, 8.353702902100309838957456786116, 9.017230424162639652057785531750, 10.32826095119858081994112276278, 11.75001281908643390669766199313, 12.28673332430732504875328307808
