| L(s) = 1 | + (−1.48 + 2.08i)2-s + (0.235 − 0.971i)3-s + (−1.47 − 4.27i)4-s + (0.193 − 4.07i)5-s + (1.67 + 1.93i)6-s + (−2.46 − 0.949i)7-s + (6.18 + 1.81i)8-s + (−0.888 − 0.458i)9-s + (8.18 + 6.43i)10-s + (−0.448 − 0.629i)11-s + (−4.50 + 0.430i)12-s + (−0.607 + 4.22i)13-s + (5.63 − 3.73i)14-s + (−3.91 − 1.14i)15-s + (−5.83 + 4.58i)16-s + (−4.50 + 0.867i)17-s + ⋯ |
| L(s) = 1 | + (−1.04 + 1.47i)2-s + (0.136 − 0.561i)3-s + (−0.739 − 2.13i)4-s + (0.0867 − 1.82i)5-s + (0.682 + 0.788i)6-s + (−0.933 − 0.358i)7-s + (2.18 + 0.642i)8-s + (−0.296 − 0.152i)9-s + (2.58 + 2.03i)10-s + (−0.135 − 0.189i)11-s + (−1.30 + 0.124i)12-s + (−0.168 + 1.17i)13-s + (1.50 − 0.997i)14-s + (−1.00 − 0.296i)15-s + (−1.45 + 1.14i)16-s + (−1.09 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.101500 - 0.236403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.101500 - 0.236403i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (2.46 + 0.949i)T \) |
| 23 | \( 1 + (4.57 + 1.43i)T \) |
| good | 2 | \( 1 + (1.48 - 2.08i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (-0.193 + 4.07i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (0.448 + 0.629i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.607 - 4.22i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (4.50 - 0.867i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-7.00 - 1.35i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (3.32 + 3.83i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (3.55 - 3.39i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-2.25 - 1.16i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-4.44 - 2.85i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (2.69 - 0.791i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (5.72 + 9.91i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.79 - 2.31i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (2.04 + 1.60i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (-0.913 - 3.76i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (-1.16 - 0.111i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (2.50 + 5.49i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (3.26 + 9.43i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (0.986 - 0.395i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (8.76 - 5.63i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-1.62 - 1.54i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (4.68 + 3.00i)T + (40.2 + 88.2i)T^{2} \) |
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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
−9.966142008844685689789982907898, −9.362341820339664763567290154029, −8.791934579744026738584591554277, −7.979536094072258309458165559731, −7.11845596319129956823608757299, −6.20505291015185351030554785315, −5.37696171974211000067651685843, −4.22192794923293110549878110710, −1.56823912079650428962153110307, −0.21443790130244735378352155553,
2.40526519767480092721315067509, 3.03342404971786073790026777622, 3.77418053000029888590129843232, 5.73526356176774492547900032340, 7.09999490548142475314466153720, 7.86574628487177502199380684368, 9.206469969137046091916265531606, 9.776435181328760608480532557166, 10.34443288050326009532976640171, 11.10014399664174513226153546592
