L(s) = 1 | + (−1.05 + 2.19i)2-s + (1.37 − 1.05i)3-s + (−2.44 − 3.06i)4-s + (−3.22 − 0.994i)5-s + (0.871 + 4.12i)6-s + (−0.531 + 2.59i)7-s + (4.56 − 1.04i)8-s + (0.761 − 2.90i)9-s + (5.58 − 6.02i)10-s + (2.38 − 0.178i)11-s + (−6.59 − 1.61i)12-s + (5.81 − 0.435i)13-s + (−5.12 − 3.90i)14-s + (−5.47 + 2.04i)15-s + (−0.789 + 3.45i)16-s + (−1.47 − 3.74i)17-s + ⋯ |
L(s) = 1 | + (−0.746 + 1.55i)2-s + (0.791 − 0.610i)3-s + (−1.22 − 1.53i)4-s + (−1.44 − 0.444i)5-s + (0.355 + 1.68i)6-s + (−0.200 + 0.979i)7-s + (1.61 − 0.368i)8-s + (0.253 − 0.967i)9-s + (1.76 − 1.90i)10-s + (0.720 − 0.0539i)11-s + (−1.90 − 0.467i)12-s + (1.61 − 0.120i)13-s + (−1.36 − 1.04i)14-s + (−1.41 + 0.528i)15-s + (−0.197 + 0.864i)16-s + (−0.356 − 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.890862 + 0.263433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.890862 + 0.263433i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.37 + 1.05i)T \) |
| 7 | \( 1 + (0.531 - 2.59i)T \) |
good | 2 | \( 1 + (1.05 - 2.19i)T + (-1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (3.22 + 0.994i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-2.38 + 0.178i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-5.81 + 0.435i)T + (12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (1.47 + 3.74i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-3.32 + 1.92i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.156 - 1.03i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-5.17 + 2.02i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + 2.94iT - 31T^{2} \) |
| 37 | \( 1 + (-11.0 - 1.65i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (3.77 - 3.50i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (2.94 + 2.73i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (2.68 + 1.29i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-1.05 - 7.02i)T + (-50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (0.112 - 0.493i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (5.84 + 4.65i)T + (13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 1.32T + 67T^{2} \) |
| 71 | \( 1 + (-8.15 + 6.50i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-9.25 - 0.693i)T + (72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 + (-1.27 + 17.0i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (4.11 + 2.80i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (4.19 + 2.42i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.43871408003442400361532528307, −9.538988644228209387364703157978, −8.965003591236453422347457454992, −8.307521090412450153593726047613, −7.76517449458812545923241791504, −6.75641798135345446882619597039, −6.01834247697709697530241920167, −4.57729552144606486044886503164, −3.24126062472131647148310986474, −0.867993191325580831737400371075,
1.27790858468826200793248452721, 3.11663546755621013920314262771, 3.80842687127078923762820885169, 4.22789105205719577524150107315, 6.73863967126059369168432322333, 8.031526497529015735017363415471, 8.388687805824409731418922759273, 9.422488274334711155989184349769, 10.33408354352425068410354420368, 10.98696124740679669933008857151