L(s) = 1 | + (−2 + i)5-s − i·7-s + 4·11-s − 2i·13-s − 2i·17-s − 2·19-s + 6i·23-s + (3 − 4i)25-s + 6·29-s − 6·31-s + (1 + 2i)35-s − 4i·37-s − 4i·43-s + 4i·47-s − 49-s + ⋯ |
L(s) = 1 | + (−0.894 + 0.447i)5-s − 0.377i·7-s + 1.20·11-s − 0.554i·13-s − 0.485i·17-s − 0.458·19-s + 1.25i·23-s + (0.600 − 0.800i)25-s + 1.11·29-s − 1.07·31-s + (0.169 + 0.338i)35-s − 0.657i·37-s − 0.609i·43-s + 0.583i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.356704333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.356704333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.913623775180740305041447614234, −7.44405477547172090249095034002, −6.76859367155054717686707958605, −6.08773216882187887150194051471, −5.10362541349360466206183278186, −4.22233960846044003554235007887, −3.63591265806906311126248800952, −2.91056700258392534705339202068, −1.62384918158256139771304552022, −0.45365333101092635130843334355,
0.953376686207962207846543021984, 1.98741969506551593509163250878, 3.14272197517303297972975012282, 4.03796273095759626209845775860, 4.48843887522150546993028800139, 5.36239487320947567970015956367, 6.54675306222340887426383423112, 6.64294914694382344016972255277, 7.81589703406006464654024524562, 8.383949834434229405597320807512