| L(s) = 1 | + (1.24 − 0.671i)2-s + (−0.528 + 1.64i)3-s + (1.09 − 1.67i)4-s + (0.546 + 0.651i)5-s + (0.448 + 2.40i)6-s + (0.348 + 0.958i)7-s + (0.245 − 2.81i)8-s + (−2.44 − 1.74i)9-s + (1.11 + 0.443i)10-s + (−1.57 − 1.31i)11-s + (2.17 + 2.69i)12-s + (−0.374 − 2.12i)13-s + (1.07 + 0.958i)14-s + (−1.36 + 0.556i)15-s + (−1.58 − 3.67i)16-s + (−6.34 + 3.66i)17-s + ⋯ |
| L(s) = 1 | + (0.880 − 0.474i)2-s + (−0.305 + 0.952i)3-s + (0.549 − 0.835i)4-s + (0.244 + 0.291i)5-s + (0.183 + 0.983i)6-s + (0.131 + 0.362i)7-s + (0.0867 − 0.996i)8-s + (−0.813 − 0.581i)9-s + (0.353 + 0.140i)10-s + (−0.474 − 0.397i)11-s + (0.627 + 0.778i)12-s + (−0.103 − 0.588i)13-s + (0.288 + 0.256i)14-s + (−0.352 + 0.143i)15-s + (−0.396 − 0.918i)16-s + (−1.53 + 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 1.88e-6i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 1.88e-6i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47004 - 1.38587\times10^{-6}i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47004 - 1.38587\times10^{-6}i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.671i)T \) |
| 3 | \( 1 + (0.528 - 1.64i)T \) |
| good | 5 | \( 1 + (-0.546 - 0.651i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.348 - 0.958i)T + (-5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.57 + 1.31i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.374 + 2.12i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (6.34 - 3.66i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.39 - 2.53i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.20 + 1.89i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.64 - 0.642i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.60 - 4.40i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-5.33 - 9.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.00 + 0.705i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.51 + 4.19i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.10 - 1.49i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 5.55iT - 53T^{2} \) |
| 59 | \( 1 + (-9.24 + 7.75i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.502 - 0.182i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (6.40 - 1.12i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.86 - 10.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.769 - 1.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (14.2 + 2.51i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.916 - 5.19i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.41 + 1.96i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.35 - 2.81i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.81293177280501893758814619726, −12.58398650592077073864144991260, −11.57278226653597035620149513850, −10.60254652583919161507293387364, −9.958068199915058490888597553551, −8.459943370244535886951554254156, −6.40152637967995900704605140926, −5.47017419014752448571477560136, −4.21707858110427999263820311373, −2.75017104370876936521867825273,
2.36846207115699871608553976169, 4.53434170724432607718414476612, 5.72451100997826164330831815652, 6.98288197229556299734439983023, 7.70453492180543227770048820287, 9.184152808361036385072394695747, 11.11198480464624084521507157548, 11.80519935893838643668939647432, 13.00641041915761242244761263122, 13.53266840991587712510262408226
