| 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
−10.59837516679449440468517723239, −10.26195037681339935084383777292, −9.137844859464752770868880039115, −7.81786435819478119715401406632, −7.22713897788024614043562672894, −6.51639648193960193598126843529, −4.48517031145122238252388701270, −3.03812106394253450483133435401, −2.49057663661399946279777840675, −0.10407415054866883791787890786,
2.20640030481755034035819807759, 4.34949790943518841379824393063, 5.20736272664810772409989912973, 5.92688475847800764072969802171, 7.51332522378606291339225202019, 8.291086862866747304857894948439, 9.066734396897628180406116211632, 9.795621735716508282128817193038, 10.44624815188984628372251193099, 12.15987522709681753512181436315
