| L(s) = 1 | + (−1.50 − 1.31i)2-s + (1.09 − 1.33i)3-s + (0.516 + 3.96i)4-s + (−0.199 − 0.658i)5-s + (−3.41 + 0.561i)6-s + (2.05 − 10.3i)7-s + (4.45 − 6.64i)8-s + (−0.585 − 2.94i)9-s + (−0.568 + 1.25i)10-s + (10.3 + 1.01i)11-s + (5.87 + 3.66i)12-s + (2.71 + 0.822i)13-s + (−16.7 + 12.8i)14-s + (−1.10 − 0.455i)15-s + (−15.4 + 4.09i)16-s + (8.04 + 19.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.751 − 0.659i)2-s + (0.366 − 0.446i)3-s + (0.129 + 0.991i)4-s + (−0.0399 − 0.131i)5-s + (−0.569 + 0.0936i)6-s + (0.293 − 1.47i)7-s + (0.557 − 0.830i)8-s + (−0.0650 − 0.326i)9-s + (−0.0568 + 0.125i)10-s + (0.938 + 0.0924i)11-s + (0.489 + 0.305i)12-s + (0.208 + 0.0632i)13-s + (−1.19 + 0.915i)14-s + (−0.0733 − 0.0303i)15-s + (−0.966 + 0.256i)16-s + (0.473 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.699 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.505102 - 1.20010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.505102 - 1.20010i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.50 + 1.31i)T \) |
| 3 | \( 1 + (-1.09 + 1.33i)T \) |
| good | 5 | \( 1 + (0.199 + 0.658i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-2.05 + 10.3i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-10.3 - 1.01i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-2.71 - 0.822i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-8.04 - 19.4i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-7.43 + 3.97i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (30.3 + 20.2i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (24.4 - 2.40i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-32.6 + 32.6i)T - 961iT^{2} \) |
| 37 | \( 1 + (12.4 + 6.67i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (56.5 + 37.7i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (37.2 + 45.3i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-16.7 - 40.5i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-24.2 - 2.39i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-9.29 - 30.6i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-8.24 + 10.0i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-18.8 - 15.4i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (88.3 + 17.5i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (45.1 - 8.98i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-45.4 + 109. i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-71.5 - 133. i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (20.1 + 30.0i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-108. - 108. i)T + 9.40e3iT^{2} \) |
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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
−10.55683423898982993530795110510, −10.07507510842056931437844392707, −8.865424351884949451168437989215, −8.108751583425317972484057422416, −7.27564182462800180224515413194, −6.38820803587579081725459528386, −4.28260698373733150099804309019, −3.59850717032531629691258484070, −1.87921772866095080982600898042, −0.75651175193008916098536442505,
1.66360936223917665030059849519, 3.16608180441760474925720658912, 4.92920256618338968485149103139, 5.72649621168669355332833890258, 6.81419159904664018080512095428, 7.987293989080428261765105690733, 8.751759031581304906956773580669, 9.439321756333830452590597357442, 10.16553558338127349932755302572, 11.59095933821192303604608195355
