L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·7-s + 0.999i·8-s − i·9-s + (−0.366 − 0.366i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (0.5 + 0.133i)22-s + 1.73i·23-s + (−0.866 − 0.499i)28-s + (1.36 − 1.36i)29-s + (0.866 + 0.499i)32-s + (−0.866 − 0.499i)36-s + (−1.36 − 1.36i)37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·7-s + 0.999i·8-s − i·9-s + (−0.366 − 0.366i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (0.5 + 0.133i)22-s + 1.73i·23-s + (−0.866 − 0.499i)28-s + (1.36 − 1.36i)29-s + (0.866 + 0.499i)32-s + (−0.866 − 0.499i)36-s + (−1.36 − 1.36i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6466309152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6466309152\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - 1.73iT - T^{2} \) |
| 29 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−8.729436048047735007815003192827, −8.062372402780433421342588980902, −7.31344521116772882698244338058, −6.72533956718400864574890997197, −5.94937502898975064133409873956, −5.14718205787458772956750077918, −4.01019231017421815406678662719, −3.12421150807017279841046489956, −1.71848451958427870153042364889, −0.55142744348754855204443520108,
1.54447278205945325016081572795, 2.50814897000018762292059693634, 3.08041788437319259167255478673, 4.52969969533071694394861849422, 5.14938842272686044754710811287, 6.35028974103452290404821346693, 6.98452658822626605182077319640, 7.974935195900551947262110962917, 8.497253431660556057892607757478, 8.977737679307837605001291854263