L(s) = 1 | + (1.82 + 0.820i)2-s + (−0.374 + 0.155i)3-s + (2.65 + 2.99i)4-s + (−7.60 − 3.15i)5-s + (−0.811 − 0.0241i)6-s + (6.84 − 6.84i)7-s + (2.39 + 7.63i)8-s + (−6.24 + 6.24i)9-s + (−11.2 − 11.9i)10-s + (−2.23 − 0.927i)11-s + (−1.46 − 0.709i)12-s + (1.40 − 0.583i)13-s + (18.0 − 6.86i)14-s + 3.34·15-s + (−1.90 + 15.8i)16-s − 2.67i·17-s + ⋯ |
L(s) = 1 | + (0.912 + 0.410i)2-s + (−0.124 + 0.0517i)3-s + (0.663 + 0.747i)4-s + (−1.52 − 0.630i)5-s + (−0.135 − 0.00402i)6-s + (0.977 − 0.977i)7-s + (0.298 + 0.954i)8-s + (−0.694 + 0.694i)9-s + (−1.12 − 1.19i)10-s + (−0.203 − 0.0842i)11-s + (−0.121 − 0.0591i)12-s + (0.108 − 0.0449i)13-s + (1.29 − 0.490i)14-s + 0.222·15-s + (−0.118 + 0.992i)16-s − 0.157i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29775 + 0.290332i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29775 + 0.290332i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.82 - 0.820i)T \) |
good | 3 | \( 1 + (0.374 - 0.155i)T + (6.36 - 6.36i)T^{2} \) |
| 5 | \( 1 + (7.60 + 3.15i)T + (17.6 + 17.6i)T^{2} \) |
| 7 | \( 1 + (-6.84 + 6.84i)T - 49iT^{2} \) |
| 11 | \( 1 + (2.23 + 0.927i)T + (85.5 + 85.5i)T^{2} \) |
| 13 | \( 1 + (-1.40 + 0.583i)T + (119. - 119. i)T^{2} \) |
| 17 | \( 1 + 2.67iT - 289T^{2} \) |
| 19 | \( 1 + (-5.38 - 13.0i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-18.8 - 18.8i)T + 529iT^{2} \) |
| 29 | \( 1 + (10.0 + 24.2i)T + (-594. + 594. i)T^{2} \) |
| 31 | \( 1 + 47.5iT - 961T^{2} \) |
| 37 | \( 1 + (28.2 + 11.7i)T + (968. + 968. i)T^{2} \) |
| 41 | \( 1 + (-6.93 + 6.93i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-8.48 - 3.51i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + 67.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (10.5 - 25.3i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-27.9 + 67.4i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-31.5 - 76.2i)T + (-2.63e3 + 2.63e3i)T^{2} \) |
| 67 | \( 1 + (-90.1 + 37.3i)T + (3.17e3 - 3.17e3i)T^{2} \) |
| 71 | \( 1 + (-1.98 + 1.98i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (55.5 - 55.5i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 10.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-34.1 - 82.5i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (16.1 + 16.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + 62.6T + 9.40e3T^{2} \) |
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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
−16.51703672695641461856067869178, −15.47817754889197860511584546939, −14.32475821174384511219469963090, −13.16435875337652486536638739612, −11.66811875713370374678253173875, −11.10163880389321867750127998990, −8.162786719097835410949965745040, −7.55796276594583304214552525377, −5.17096626389097321393372973128, −3.94138188045281269372933294009,
3.17621368502709902051459804902, 5.01369407875409294817245133075, 6.87209929754628463303825851602, 8.571983839332988318348024922484, 10.93784320107748705480382722311, 11.62528509490909826198470621404, 12.48182422396129836640789984178, 14.48720441374664396857983013932, 15.00773193485705525265960067312, 15.97624096873543737382770955214