L(s) = 1 | + (0.679 − 1.96i)2-s + (0.458 + 0.888i)3-s + (−1.81 − 1.42i)4-s + (−3.97 + 0.379i)5-s + (2.05 − 0.295i)6-s + (0.921 − 2.48i)7-s + (−0.546 + 0.351i)8-s + (−0.580 + 0.814i)9-s + (−1.95 + 8.05i)10-s + (−3.49 + 1.20i)11-s + (0.437 − 2.27i)12-s + (−1.42 − 4.84i)13-s + (−4.24 − 3.49i)14-s + (−2.15 − 3.35i)15-s + (−0.772 − 3.18i)16-s + (−3.63 − 1.45i)17-s + ⋯ |
L(s) = 1 | + (0.480 − 1.38i)2-s + (0.264 + 0.513i)3-s + (−0.908 − 0.714i)4-s + (−1.77 + 0.169i)5-s + (0.839 − 0.120i)6-s + (0.348 − 0.937i)7-s + (−0.193 + 0.124i)8-s + (−0.193 + 0.271i)9-s + (−0.617 + 2.54i)10-s + (−1.05 + 0.364i)11-s + (0.126 − 0.655i)12-s + (−0.394 − 1.34i)13-s + (−1.13 − 0.933i)14-s + (−0.556 − 0.866i)15-s + (−0.193 − 0.796i)16-s + (−0.881 − 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.147229 + 0.794664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.147229 + 0.794664i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.921 + 2.48i)T \) |
| 23 | \( 1 + (-4.30 + 2.11i)T \) |
good | 2 | \( 1 + (-0.679 + 1.96i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (3.97 - 0.379i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (3.49 - 1.20i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (1.42 + 4.84i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (3.63 + 1.45i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (6.01 - 2.40i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.765 - 5.32i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.45 + 0.116i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-0.999 - 0.711i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-1.81 - 0.830i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.70 + 8.87i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-3.43 + 1.98i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.54 - 1.62i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (6.82 + 1.65i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-3.80 - 1.96i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (-0.334 - 1.73i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (8.56 + 9.88i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-7.91 + 10.0i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (2.88 - 3.03i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.63 + 3.57i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.403 - 8.47i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (-1.21 + 2.66i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.71339568365979410922433696434, −10.26697326788679592924203776709, −8.717791775661332835319271939202, −7.80913308195210959043487178791, −7.15207049569203169016586794030, −4.93339587566317463672844869390, −4.38743621651206586539263284311, −3.50513563572434513746381591671, −2.59727938889375034622227178431, −0.39266618475749447714752596051,
2.49919286376956776396876662545, 4.21436500325569975080952243460, 4.78701512615799384354402755175, 6.10876179776683262827042257590, 6.99979552168641711181435700612, 7.76230830221092872500969317970, 8.438194166543183849619881962257, 8.974116282884531250473809040944, 11.04817034520192864021556368288, 11.48583073831716687481612006742