L(s) = 1 | + (1.41 − 0.0660i)2-s − 0.496·3-s + (1.99 − 0.186i)4-s + (−2.00 − 0.987i)5-s + (−0.701 + 0.0328i)6-s + (1.55 + 1.55i)7-s + (2.80 − 0.395i)8-s − 2.75·9-s + (−2.89 − 1.26i)10-s + (−4.19 + 4.19i)11-s + (−0.988 + 0.0927i)12-s − 5.09i·13-s + (2.29 + 2.09i)14-s + (0.996 + 0.490i)15-s + (3.93 − 0.743i)16-s + (0.213 + 0.213i)17-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0467i)2-s − 0.286·3-s + (0.995 − 0.0933i)4-s + (−0.897 − 0.441i)5-s + (−0.286 + 0.0133i)6-s + (0.587 + 0.587i)7-s + (0.990 − 0.139i)8-s − 0.917·9-s + (−0.916 − 0.399i)10-s + (−1.26 + 1.26i)11-s + (−0.285 + 0.0267i)12-s − 1.41i·13-s + (0.614 + 0.559i)14-s + (0.257 + 0.126i)15-s + (0.982 − 0.185i)16-s + (0.0517 + 0.0517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33117 - 0.0728634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33117 - 0.0728634i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 + 0.0660i)T \) |
| 5 | \( 1 + (2.00 + 0.987i)T \) |
good | 3 | \( 1 + 0.496T + 3T^{2} \) |
| 7 | \( 1 + (-1.55 - 1.55i)T + 7iT^{2} \) |
| 11 | \( 1 + (4.19 - 4.19i)T - 11iT^{2} \) |
| 13 | \( 1 + 5.09iT - 13T^{2} \) |
| 17 | \( 1 + (-0.213 - 0.213i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.844 + 0.844i)T - 19iT^{2} \) |
| 23 | \( 1 + (-1.70 + 1.70i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.24 - 2.24i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.818iT - 31T^{2} \) |
| 37 | \( 1 + 5.12iT - 37T^{2} \) |
| 41 | \( 1 - 3.34iT - 41T^{2} \) |
| 43 | \( 1 - 4.49iT - 43T^{2} \) |
| 47 | \( 1 + (-4.29 + 4.29i)T - 47iT^{2} \) |
| 53 | \( 1 + 1.00T + 53T^{2} \) |
| 59 | \( 1 + (7.65 + 7.65i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.90 - 1.90i)T - 61iT^{2} \) |
| 67 | \( 1 - 11.0iT - 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + (-2.70 - 2.70i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.32T + 79T^{2} \) |
| 83 | \( 1 + 9.17T + 83T^{2} \) |
| 89 | \( 1 + 4.25T + 89T^{2} \) |
| 97 | \( 1 + (7.15 + 7.15i)T + 97iT^{2} \) |
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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
−14.59855240063187609005930318053, −12.98025049548092383137913306227, −12.37901667288690164834641086643, −11.39500172540144681204082650881, −10.42162312147607482937588967410, −8.358868483545534764774981994394, −7.42298903810641290860573678287, −5.53037620697632889151644035045, −4.79443285457722772839826610075, −2.84000252646329249803409146391,
3.06187346871391095084032398811, 4.54159301100008861327880128511, 5.93215966959188736103960503054, 7.32166178882836081414107368594, 8.357535148674643010251516203813, 10.71534314597435587853256524963, 11.26328412528775299033848077462, 12.07366657820142949147814534885, 13.69297980386297648504775100503, 14.15685738139674212737010779481