L(s) = 1 | + (3 + 1.73i)5-s + (0.5 + 2.59i)7-s + (3 − 1.73i)11-s − 5.19i·13-s + (6 − 3.46i)17-s + (−3.5 + 6.06i)19-s + (3.5 + 6.06i)25-s + (−2.5 − 4.33i)31-s + (−3 + 8.66i)35-s + (−0.5 + 0.866i)37-s + 10.3i·41-s − 1.73i·43-s + (−3 + 5.19i)47-s + (−6.5 + 2.59i)49-s + 12·55-s + ⋯ |
L(s) = 1 | + (1.34 + 0.774i)5-s + (0.188 + 0.981i)7-s + (0.904 − 0.522i)11-s − 1.44i·13-s + (1.45 − 0.840i)17-s + (−0.802 + 1.39i)19-s + (0.700 + 1.21i)25-s + (−0.449 − 0.777i)31-s + (−0.507 + 1.46i)35-s + (−0.0821 + 0.142i)37-s + 1.62i·41-s − 0.264i·43-s + (−0.437 + 0.757i)47-s + (−0.928 + 0.371i)49-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.209235144\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.209235144\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (-6 + 3.46i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 0.866i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (13.5 + 7.79i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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
−9.926643816528771425453593935499, −9.460921995055035657665285988072, −8.383216908065621612979749664439, −7.62190648936649790895299040758, −6.22688278065087222181262375563, −5.95874539460402550803226175504, −5.15166699712721117624005442623, −3.45105511310818678399443460005, −2.67290374323381343025240812578, −1.47271968346938712348794827883,
1.24848984411413032704653694327, 2.02841478795837421035871178007, 3.81087117394956968277353124733, 4.61482375082979772642949919702, 5.53337107360766385145487256840, 6.60451195973772045311178607638, 7.11363521738947374123338657465, 8.476465864121345578363289494351, 9.155022842141244979940517030927, 9.800331275349243628524798112220