| L(s) = 1 | + 1.41i·2-s + (−1.59 + 2.54i)3-s − 2.00·4-s + (5.46 + 3.15i)5-s + (−3.59 − 2.25i)6-s + (−3.23 + 6.20i)7-s − 2.82i·8-s + (−3.93 − 8.09i)9-s + (−4.45 + 7.72i)10-s + (−4.22 + 2.43i)11-s + (3.18 − 5.08i)12-s + (−1.62 − 2.81i)13-s + (−8.77 − 4.57i)14-s + (−16.7 + 8.87i)15-s + 4.00·16-s + (−17.6 − 10.2i)17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.530 + 0.847i)3-s − 0.500·4-s + (1.09 + 0.630i)5-s + (−0.599 − 0.375i)6-s + (−0.462 + 0.886i)7-s − 0.353i·8-s + (−0.437 − 0.899i)9-s + (−0.445 + 0.772i)10-s + (−0.383 + 0.221i)11-s + (0.265 − 0.423i)12-s + (−0.124 − 0.216i)13-s + (−0.627 − 0.326i)14-s + (−1.11 + 0.591i)15-s + 0.250·16-s + (−1.04 − 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.144319 + 1.07991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144319 + 1.07991i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (1.59 - 2.54i)T \) |
| 7 | \( 1 + (3.23 - 6.20i)T \) |
| good | 5 | \( 1 + (-5.46 - 3.15i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (4.22 - 2.43i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1.62 + 2.81i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (17.6 + 10.2i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.7 - 25.5i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (5.47 + 3.16i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-23.6 - 13.6i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 46.9T + 961T^{2} \) |
| 37 | \( 1 + (-25.7 - 44.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-29.7 + 17.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (34.9 - 60.4i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 - 12.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-2.78 - 1.60i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + 105. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 67.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 26.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 59.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-17.7 + 30.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + 72.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + (110. + 63.5i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-35.7 + 20.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (73.4 - 127. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.87511667344043857722902662633, −12.65568808786091327321053625146, −11.48180790280511913783919308398, −9.997506661164659011505670800626, −9.763370675089134283541098190240, −8.405022204328504592373918700479, −6.58856281137187610702729254750, −5.91517632842973522050113284267, −4.83399786260513788178103479188, −2.88886080578592769060867399720,
0.821973384451414986897933151364, 2.41590116417062907294790049927, 4.60927858628776780211633483814, 5.88223932197745903316964046690, 7.04670227298995403880272275601, 8.533286556066948821156180734035, 9.717579311771752299668225797595, 10.66844140321992986293254859422, 11.68181553902989749191372390368, 12.88557131087571692047872773602
