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 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−9.800331275349243628524798112220, −9.155022842141244979940517030927, −8.476465864121345578363289494351, −7.11363521738947374123338657465, −6.60451195973772045311178607638, −5.53337107360766385145487256840, −4.61482375082979772642949919702, −3.81087117394956968277353124733, −2.02841478795837421035871178007, −1.24848984411413032704653694327,
1.47271968346938712348794827883, 2.67290374323381343025240812578, 3.45105511310818678399443460005, 5.15166699712721117624005442623, 5.95874539460402550803226175504, 6.22688278065087222181262375563, 7.62190648936649790895299040758, 8.383216908065621612979749664439, 9.460921995055035657665285988072, 9.926643816528771425453593935499