L(s) = 1 | + (0.999 − 2.54i)2-s + (−0.421 − 1.68i)3-s + (−4.01 − 3.72i)4-s + (2.24 − 1.08i)5-s + (−4.69 − 0.606i)6-s + (1.30 + 2.30i)7-s + (−8.57 + 4.13i)8-s + (−2.64 + 1.41i)9-s + (−0.510 − 6.80i)10-s + (−1.80 − 2.25i)11-s + (−4.57 + 8.32i)12-s + (0.380 − 0.970i)13-s + (7.16 − 1.01i)14-s + (−2.76 − 3.32i)15-s + (1.12 + 15.0i)16-s + (2.08 − 1.93i)17-s + ⋯ |
L(s) = 1 | + (0.706 − 1.80i)2-s + (−0.243 − 0.969i)3-s + (−2.00 − 1.86i)4-s + (1.00 − 0.484i)5-s + (−1.91 − 0.247i)6-s + (0.491 + 0.870i)7-s + (−3.03 + 1.46i)8-s + (−0.881 + 0.471i)9-s + (−0.161 − 2.15i)10-s + (−0.543 − 0.681i)11-s + (−1.31 + 2.40i)12-s + (0.105 − 0.269i)13-s + (1.91 − 0.270i)14-s + (−0.714 − 0.857i)15-s + (0.281 + 3.75i)16-s + (0.506 − 0.470i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.719961 + 1.60009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.719961 + 1.60009i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.421 + 1.68i)T \) |
| 7 | \( 1 + (-1.30 - 2.30i)T \) |
good | 2 | \( 1 + (-0.999 + 2.54i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.24 + 1.08i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (1.80 + 2.25i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.380 + 0.970i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-2.08 + 1.93i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.06 - 3.57i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.43 + 6.27i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-8.99 - 2.77i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-3.80 + 6.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.99 - 0.616i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (5.51 + 3.76i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (0.626 - 0.426i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-0.937 + 2.38i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-9.05 + 2.79i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (11.5 - 7.87i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (1.81 - 1.68i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.605 + 1.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.92 - 8.43i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (2.53 + 0.382i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (1.42 + 2.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.443 - 1.13i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-3.75 - 9.57i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (4.19 - 7.27i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.72371185338949873783737533357, −10.05802346621551785085990133121, −8.851370253378130060345829467917, −8.302411951088048358795523980136, −6.13663451883310977849131231342, −5.56932908198527541880397900722, −4.75434283016909413066319809701, −2.90354854964363163256231312220, −2.16044800261853905045716443797, −0.981565955714663536031661813480,
3.18046708581227060037860320247, 4.46848785338909943989575728849, 5.01344214551903736474554095454, 6.06993197613383229799939312192, 6.78713779482384321744911427125, 7.81188954515679352145892298889, 8.785209257668983474194063454590, 9.820076842273980978401840203458, 10.45970786991832048130804467006, 11.81628599994551629411476386594