| L(s) = 1 | + (−1.40 − 0.125i)2-s + (1.89 + 0.786i)3-s + (1.96 + 0.354i)4-s + (2.17 − 1.10i)5-s + (−2.57 − 1.34i)6-s + (0.151 + 0.247i)7-s + (−2.72 − 0.746i)8-s + (0.867 + 0.867i)9-s + (−3.20 + 1.28i)10-s + (−2.25 − 0.177i)11-s + (3.46 + 2.22i)12-s + (−0.172 − 0.717i)13-s + (−0.182 − 0.368i)14-s + (5.00 − 0.393i)15-s + (3.74 + 1.39i)16-s + (−3.59 + 3.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0889i)2-s + (1.09 + 0.454i)3-s + (0.984 + 0.177i)4-s + (0.972 − 0.495i)5-s + (−1.05 − 0.550i)6-s + (0.0574 + 0.0937i)7-s + (−0.964 − 0.264i)8-s + (0.289 + 0.289i)9-s + (−1.01 + 0.407i)10-s + (−0.680 − 0.0535i)11-s + (0.998 + 0.641i)12-s + (−0.0477 − 0.198i)13-s + (−0.0488 − 0.0984i)14-s + (1.29 − 0.101i)15-s + (0.937 + 0.348i)16-s + (−0.872 + 0.744i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14025 + 0.0502349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14025 + 0.0502349i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.125i)T \) |
| 41 | \( 1 + (5.78 + 2.75i)T \) |
| good | 3 | \( 1 + (-1.89 - 0.786i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-2.17 + 1.10i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.151 - 0.247i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (2.25 + 0.177i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.172 + 0.717i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (3.59 - 3.07i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.49 - 0.838i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-2.94 + 2.14i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.88 + 3.31i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-2.10 - 6.49i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.56 - 10.9i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (9.67 + 1.53i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.192 + 0.314i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-5.01 + 5.87i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (3.42 + 4.71i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.82 + 1.55i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.538 + 6.84i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (0.568 - 7.22i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (5.96 - 5.96i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.14 + 12.4i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 17.9iT - 83T^{2} \) |
| 89 | \( 1 + (2.99 - 1.83i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (0.792 + 10.0i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−13.00304010913383671087580515102, −11.68454791714284629981182859161, −10.31824214581444261483486413419, −9.766948937410236553236617387828, −8.720941477099151441974330263311, −8.259981632708987208311150138749, −6.73393395651711032064489067610, −5.30276741392595096145793749282, −3.27870901787258349318987097330, −1.97166482739704673013645615242,
2.00654663476093327734553335028, 2.94530147199914927146235115137, 5.54652465933617030927454114083, 6.95341866942774111286429799380, 7.64978351752198741141648994979, 8.871859718261248610268236923565, 9.526647597441014373960866753265, 10.56996003214234465476011871219, 11.57400814261493607733052072459, 13.13579783747430469759541083752
