L(s) = 1 | + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−1.22 − 0.707i)5-s + 2.82i·8-s + (−0.999 − 1.73i)10-s + (−6.12 + 3.53i)11-s − 15·13-s + (−2.00 + 3.46i)16-s + (9.79 − 5.65i)17-s + (−6.5 + 11.2i)19-s − 2.82i·20-s − 10·22-s + (−19.5 − 11.3i)23-s + (−11.5 − 19.9i)25-s + (−18.3 − 10.6i)26-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.244 − 0.141i)5-s + 0.353i·8-s + (−0.0999 − 0.173i)10-s + (−0.556 + 0.321i)11-s − 1.15·13-s + (−0.125 + 0.216i)16-s + (0.576 − 0.332i)17-s + (−0.342 + 0.592i)19-s − 0.141i·20-s − 0.454·22-s + (−0.851 − 0.491i)23-s + (−0.460 − 0.796i)25-s + (−0.706 − 0.407i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1038496787\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1038496787\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.22 + 0.707i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (6.12 - 3.53i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 15T + 169T^{2} \) |
| 17 | \( 1 + (-9.79 + 5.65i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (6.5 - 11.2i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (19.5 + 11.3i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 22.6iT - 841T^{2} \) |
| 31 | \( 1 + (-1.5 - 2.59i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (8.5 - 14.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 80.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 85T + 1.84e3T^{2} \) |
| 47 | \( 1 + (62.4 + 36.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-29.3 + 16.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-78.3 + 45.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (36 - 62.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (21.5 + 37.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 52.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (47.5 + 82.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (34.5 - 59.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 60.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (117. + 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 16T + 9.40e3T^{2} \) |
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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
−10.08063279357178973993698149643, −9.954488048754235189239654494041, −8.355558041521206775448678667309, −7.922402638348804819833188926819, −6.96020437467112616414894780102, −6.06528913831099287802504792570, −5.03566780952908430956341602123, −4.36993682790679797714824942183, −3.15740220795478213568690229849, −2.04668376235492404444613472322,
0.02489218043299588480738138522, 1.82232010916850426049726352961, 2.97299494008525809463075989582, 3.89692384784599028842702539867, 5.04047642322960472550596782251, 5.68581790852428517463482733928, 6.89791882722364388931751900275, 7.60852740509002693088875553001, 8.611778575638269589231023269129, 9.717571787949153523892278698336