| L(s) = 1 | + (−0.870 + 1.11i)2-s + (0.290 + 0.956i)3-s + (−0.484 − 1.94i)4-s + (−0.328 − 3.33i)5-s + (−1.31 − 0.509i)6-s + (−1.53 + 2.29i)7-s + (2.58 + 1.15i)8-s + (−0.831 + 0.555i)9-s + (4.00 + 2.53i)10-s + (−2.66 + 1.42i)11-s + (1.71 − 1.02i)12-s + (−6.16 − 0.607i)13-s + (−1.22 − 3.70i)14-s + (3.09 − 1.28i)15-s + (−3.53 + 1.87i)16-s + (1.52 + 0.633i)17-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.788i)2-s + (0.167 + 0.552i)3-s + (−0.242 − 0.970i)4-s + (−0.146 − 1.49i)5-s + (−0.538 − 0.208i)6-s + (−0.579 + 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.277 + 0.185i)9-s + (1.26 + 0.802i)10-s + (−0.803 + 0.429i)11-s + (0.495 − 0.296i)12-s + (−1.70 − 0.168i)13-s + (−0.326 − 0.989i)14-s + (0.799 − 0.331i)15-s + (−0.882 + 0.469i)16-s + (0.370 + 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0360648 - 0.137817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0360648 - 0.137817i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.870 - 1.11i)T \) |
| 3 | \( 1 + (-0.290 - 0.956i)T \) |
| good | 5 | \( 1 + (0.328 + 3.33i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (1.53 - 2.29i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (2.66 - 1.42i)T + (6.11 - 9.14i)T^{2} \) |
| 13 | \( 1 + (6.16 + 0.607i)T + (12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 0.633i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (6.31 - 5.18i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (0.448 - 2.25i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.20 + 2.24i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (-0.352 + 0.352i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.37 + 1.67i)T + (-7.21 - 36.2i)T^{2} \) |
| 41 | \( 1 + (-0.719 - 0.143i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.498 + 1.64i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-5.00 + 12.0i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.75 - 3.28i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (3.13 - 0.308i)T + (57.8 - 11.5i)T^{2} \) |
| 61 | \( 1 + (9.72 - 2.94i)T + (50.7 - 33.8i)T^{2} \) |
| 67 | \( 1 + (-1.59 + 0.484i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (11.9 + 7.97i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (2.78 + 4.16i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-5.23 - 12.6i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (0.835 + 1.01i)T + (-16.1 + 81.4i)T^{2} \) |
| 89 | \( 1 + (-3.31 - 16.6i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-3.58 + 3.58i)T - 97iT^{2} \) |
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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
−12.15987522709681753512181436315, −10.44624815188984628372251193099, −9.795621735716508282128817193038, −9.066734396897628180406116211632, −8.291086862866747304857894948439, −7.51332522378606291339225202019, −5.92688475847800764072969802171, −5.20736272664810772409989912973, −4.34949790943518841379824393063, −2.20640030481755034035819807759,
0.10407415054866883791787890786, 2.49057663661399946279777840675, 3.03812106394253450483133435401, 4.48517031145122238252388701270, 6.51639648193960193598126843529, 7.22713897788024614043562672894, 7.81786435819478119715401406632, 9.137844859464752770868880039115, 10.26195037681339935084383777292, 10.59837516679449440468517723239
