L(s) = 1 | + (−1.34 + 0.359i)2-s + (−0.0625 + 0.0361i)4-s + (2.52 − 2.52i)5-s + (−1.59 + 2.11i)7-s + (2.03 − 2.03i)8-s + (−2.48 + 4.29i)10-s + (−0.0529 − 0.197i)11-s + (2.07 − 2.94i)13-s + (1.37 − 3.40i)14-s + (−1.92 + 3.33i)16-s + (1.13 + 1.97i)17-s + (1.50 + 0.402i)19-s + (−0.0668 + 0.249i)20-s + (0.142 + 0.246i)22-s + (−2.59 − 1.49i)23-s + ⋯ |
L(s) = 1 | + (−0.948 + 0.254i)2-s + (−0.0312 + 0.0180i)4-s + (1.13 − 1.13i)5-s + (−0.602 + 0.798i)7-s + (0.719 − 0.719i)8-s + (−0.785 + 1.35i)10-s + (−0.0159 − 0.0596i)11-s + (0.575 − 0.817i)13-s + (0.368 − 0.910i)14-s + (−0.481 + 0.833i)16-s + (0.275 + 0.477i)17-s + (0.344 + 0.0923i)19-s + (−0.0149 + 0.0558i)20-s + (0.0302 + 0.0524i)22-s + (−0.541 − 0.312i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.898834 - 0.333379i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.898834 - 0.333379i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.59 - 2.11i)T \) |
| 13 | \( 1 + (-2.07 + 2.94i)T \) |
good | 2 | \( 1 + (1.34 - 0.359i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-2.52 + 2.52i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.0529 + 0.197i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.13 - 1.97i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.50 - 0.402i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.75 + 8.23i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.75 - 2.75i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.17 - 4.39i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.36 + 5.07i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.76 + 1.01i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (9.42 + 9.42i)T + 47iT^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + (0.510 - 1.90i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.850 + 0.491i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-15.1 + 4.06i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.00355 - 0.0132i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (2.24 + 2.24i)T + 73iT^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + (-3.37 + 3.37i)T - 83iT^{2} \) |
| 89 | \( 1 + (-11.8 + 3.17i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-7.85 - 2.10i)T + (84.0 + 48.5i)T^{2} \) |
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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
−9.936353921625557647431103456393, −9.236637434112641073101010038094, −8.504611144753661525599266213093, −8.073386475436771809315798821282, −6.61334303771091712967960498785, −5.81642967875936306560924994139, −5.02398901095145682370116032908, −3.66668006813171411767086117890, −2.09079358509766316205825549515, −0.75616036044968259542562630489,
1.27577740281959189105287388800, 2.48998698503769279748587323213, 3.69623008793300497501248614279, 5.08660654251475731235752308681, 6.23188043888729509801459934607, 6.92723761636680715242561329561, 7.77034208691616561262681301805, 9.011038018002270108906565600395, 9.634748015251463935577185733812, 10.15813041215816782065930819154