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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.966649279\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.966649279\) |
\(L(\frac{3}{2})\) |
|
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
−9.837652035559882592675033631905, −9.299750802981559027408933590020, −8.513528867911589940564655701781, −7.69510935076730065069043461413, −6.24230118974771331470176126924, −5.71716745923910514169809733608, −5.14740552790729556275239509453, −3.71641947114840228992975792657, −2.38012340947232393790553933578, −1.46824724639271500189539877841,
1.00678612609309744881587395639, 2.69059908666722207005943690410, 3.19548452408356376078755964195, 4.95335560551707424016632813275, 5.48892943691590873712993826794, 6.60243585751128372839360272086, 7.33547611781743160015597808384, 8.013045436342747691905712330580, 9.484821166444071539078997288989, 10.00618018602297004553007880282