| L(s) = 1 | + (0.961 + 1.75i)2-s + (1.09 − 1.33i)3-s + (−2.15 + 3.37i)4-s + (−1.93 − 6.37i)5-s + (3.40 + 0.639i)6-s + (−0.200 + 1.00i)7-s + (−7.98 − 0.528i)8-s + (−0.585 − 2.94i)9-s + (9.32 − 9.52i)10-s + (−18.2 − 1.79i)11-s + (2.15 + 6.58i)12-s + (11.6 + 3.54i)13-s + (−1.95 + 0.617i)14-s + (−10.6 − 4.41i)15-s + (−6.74 − 14.5i)16-s + (−9.47 − 22.8i)17-s + ⋯ |
| L(s) = 1 | + (0.480 + 0.876i)2-s + (0.366 − 0.446i)3-s + (−0.537 + 0.843i)4-s + (−0.386 − 1.27i)5-s + (0.567 + 0.106i)6-s + (−0.0286 + 0.143i)7-s + (−0.997 − 0.0660i)8-s + (−0.0650 − 0.326i)9-s + (0.932 − 0.952i)10-s + (−1.65 − 0.163i)11-s + (0.179 + 0.548i)12-s + (0.899 + 0.272i)13-s + (−0.139 + 0.0440i)14-s + (−0.710 − 0.294i)15-s + (−0.421 − 0.906i)16-s + (−0.557 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0724 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0724 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.726896 - 0.781605i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.726896 - 0.781605i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.961 - 1.75i)T \) |
| 3 | \( 1 + (-1.09 + 1.33i)T \) |
| good | 5 | \( 1 + (1.93 + 6.37i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (0.200 - 1.00i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (18.2 + 1.79i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-11.6 - 3.54i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (9.47 + 22.8i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (2.08 - 1.11i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (23.0 + 15.3i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (6.59 - 0.649i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-17.0 + 17.0i)T - 961iT^{2} \) |
| 37 | \( 1 + (50.1 + 26.8i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-59.8 - 39.9i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (30.7 + 37.5i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-20.8 - 50.4i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (17.8 + 1.76i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (14.2 + 47.0i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (10.6 - 12.9i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-87.9 - 72.1i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (86.8 + 17.2i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-121. + 24.1i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (34.6 - 83.6i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-26.4 - 49.5i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (32.3 + 48.3i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (40.7 + 40.7i)T + 9.40e3iT^{2} \) |
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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
−11.12291128652040092027245983714, −9.551497650375945507491570470188, −8.608679037295075761768015611347, −8.138371523775600018756707646998, −7.23894708113939010619714319052, −5.97013932841760636360599270497, −5.05201939891783757128956919987, −4.11361751051706061722111565756, −2.61136010907471651106135207605, −0.36485418013282506597834888706,
2.15572407091799743177614462215, 3.23722631473523298342233600819, 3.99685347347189149154921786440, 5.35929508462845066787604172557, 6.42307331145757873892551502260, 7.78191250116261133903475308962, 8.670928392007705603736065326663, 10.07962743290918694733530708709, 10.59338680109305980342027446283, 11.00063535363577240706098258540
