| L(s) = 1 | + (1.89 − 0.648i)2-s + (−1.09 + 1.33i)3-s + (3.15 − 2.45i)4-s + (0.348 + 1.14i)5-s + (−1.20 + 3.24i)6-s + (2.24 − 11.2i)7-s + (4.38 − 6.69i)8-s + (−0.585 − 2.94i)9-s + (1.40 + 1.94i)10-s + (−2.51 − 0.247i)11-s + (−0.182 + 6.92i)12-s + (−6.19 − 1.87i)13-s + (−3.07 − 22.7i)14-s + (−1.91 − 0.794i)15-s + (3.94 − 15.5i)16-s + (0.544 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (0.945 − 0.324i)2-s + (−0.366 + 0.446i)3-s + (0.789 − 0.613i)4-s + (0.0696 + 0.229i)5-s + (−0.201 + 0.540i)6-s + (0.320 − 1.61i)7-s + (0.547 − 0.836i)8-s + (−0.0650 − 0.326i)9-s + (0.140 + 0.194i)10-s + (−0.228 − 0.0224i)11-s + (−0.0152 + 0.577i)12-s + (−0.476 − 0.144i)13-s + (−0.219 − 1.62i)14-s + (−0.127 − 0.0529i)15-s + (0.246 − 0.969i)16-s + (0.0320 + 0.0773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.15276 - 1.58701i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.15276 - 1.58701i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.89 + 0.648i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-0.348 - 1.14i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-2.24 + 11.2i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (2.51 + 0.247i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (6.19 + 1.87i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-0.544 - 1.31i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-7.16 + 3.82i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-1.79 - 1.20i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (15.8 - 1.56i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-30.6 + 30.6i)T - 961iT^{2} \) |
| 37 | \( 1 + (10.1 + 5.40i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-53.5 - 35.7i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-30.4 - 37.0i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (9.37 + 22.6i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (57.0 + 5.61i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-28.9 - 95.3i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (57.7 - 70.3i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (4.33 + 3.55i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-58.0 - 11.5i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (65.5 - 13.0i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (19.9 - 48.1i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-18.6 - 34.8i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (13.5 + 20.3i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (15.9 + 15.9i)T + 9.40e3iT^{2} \) |
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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
−10.94153877473311248609377737445, −10.36251541376985436483740993266, −9.587239825451052935389361608623, −7.79040252675343035209498141073, −7.00693456299943826545518594408, −5.95432933297774386883986957598, −4.74121038303362207479748625936, −4.10576357749442635927455329284, −2.83160093478378497707390398226, −0.954045309264060625055116241795,
1.92827352971595477331442488610, 3.02650771410231439730300619857, 4.79271422830280119793497043853, 5.43737631723164874606679703079, 6.29537681710520297786248906602, 7.39152392630765549247593132928, 8.341307142037061182723662227249, 9.289778267518113256503350729870, 10.80220919927517236103671739217, 11.64088290973534287320436549456
