| L(s) = 1 | + (−0.959 − 1.03i)2-s + (0.643 + 1.60i)3-s + (−0.159 + 1.99i)4-s + (0.866 − 0.152i)5-s + (1.05 − 2.21i)6-s + (−1.10 − 0.924i)7-s + (2.22 − 1.74i)8-s + (−2.17 + 2.07i)9-s + (−0.989 − 0.753i)10-s + (6.03 + 1.06i)11-s + (−3.30 + 1.02i)12-s + (−2.10 + 5.77i)13-s + (0.0963 + 2.03i)14-s + (0.803 + 1.29i)15-s + (−3.94 − 0.635i)16-s + (0.643 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.734i)2-s + (0.371 + 0.928i)3-s + (−0.0796 + 0.996i)4-s + (0.387 − 0.0683i)5-s + (0.429 − 0.902i)6-s + (−0.416 − 0.349i)7-s + (0.786 − 0.617i)8-s + (−0.723 + 0.690i)9-s + (−0.312 − 0.238i)10-s + (1.81 + 0.320i)11-s + (−0.955 + 0.296i)12-s + (−0.582 + 1.60i)13-s + (0.0257 + 0.543i)14-s + (0.207 + 0.334i)15-s + (−0.987 − 0.158i)16-s + (0.155 + 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.980794 + 0.265381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.980794 + 0.265381i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 1.03i)T \) |
| 3 | \( 1 + (-0.643 - 1.60i)T \) |
| good | 5 | \( 1 + (-0.866 + 0.152i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.10 + 0.924i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-6.03 - 1.06i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.10 - 5.77i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.643 - 1.11i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.02 - 2.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 + 1.13i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.98 + 8.21i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.99 + 4.19i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.66 + 2.11i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.25 + 0.455i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (3.73 + 0.658i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (3.37 + 2.83i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 1.79iT - 53T^{2} \) |
| 59 | \( 1 + (-3.57 + 0.630i)T + (55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.77 - 6.88i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.22 + 6.11i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.87 + 8.44i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.531 + 0.921i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.22 + 1.53i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.85 + 5.09i)T + (-63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-4.01 + 6.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.649 - 3.68i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.82074625181788096693514393046, −11.58972368286992136073884816880, −10.05218771049778899936878188278, −9.575521289990828900618044840787, −9.007560176905868785144699351084, −7.63580235648499313334421203248, −6.37596758871356026702891146102, −4.40436253451170454037642292387, −3.66970326855608581722127293801, −1.95487161218599116108963474161,
1.19566550210197877310928162311, 3.05987954974812021142015610193, 5.37348011710445629040740568358, 6.34801514195491334576024122825, 7.15533982425462581932016733136, 8.212551970961361572025701800456, 9.182735691508304559015406194593, 9.839442252987614087247773514758, 11.30208557071390110459991281848, 12.28908120569009207802586899612
