L(s) = 1 | + (−1.30 + 0.549i)2-s + (1.62 + 0.612i)3-s + (1.39 − 1.43i)4-s + (3.01 + 2.01i)5-s + (−2.44 + 0.0924i)6-s + (−0.924 − 2.23i)7-s + (−1.03 + 2.63i)8-s + (2.25 + 1.98i)9-s + (−5.03 − 0.969i)10-s + (−0.750 − 3.77i)11-s + (3.13 − 1.46i)12-s + (−1.67 − 2.49i)13-s + (2.42 + 2.39i)14-s + (3.65 + 5.11i)15-s + (−0.0989 − 3.99i)16-s + (−4.69 + 4.69i)17-s + ⋯ |
L(s) = 1 | + (−0.921 + 0.388i)2-s + (0.935 + 0.353i)3-s + (0.698 − 0.715i)4-s + (1.34 + 0.901i)5-s + (−0.999 + 0.0377i)6-s + (−0.349 − 0.843i)7-s + (−0.365 + 0.930i)8-s + (0.750 + 0.661i)9-s + (−1.59 − 0.306i)10-s + (−0.226 − 1.13i)11-s + (0.906 − 0.422i)12-s + (−0.463 − 0.693i)13-s + (0.649 + 0.641i)14-s + (0.943 + 1.31i)15-s + (−0.0247 − 0.999i)16-s + (−1.13 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12447 + 0.446459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12447 + 0.446459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.30 - 0.549i)T \) |
| 3 | \( 1 + (-1.62 - 0.612i)T \) |
good | 5 | \( 1 + (-3.01 - 2.01i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.924 + 2.23i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.750 + 3.77i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (1.67 + 2.49i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (4.69 - 4.69i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.32 - 3.48i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.388 + 0.937i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (5.36 + 1.06i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + (1.55 + 1.03i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-3.48 - 1.44i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (0.928 + 4.66i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (8.26 + 8.26i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.23 - 0.444i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (4.00 - 5.98i)T + (-22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-0.926 - 0.184i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.204 + 1.02i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-8.71 + 3.61i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.22 - 0.923i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (3.23 + 3.23i)T + 79iT^{2} \) |
| 83 | \( 1 + (8.06 - 5.38i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-15.0 + 6.21i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 17.0iT - 97T^{2} \) |
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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
−13.11343561574761352235831482206, −10.94160653568806637158588013453, −10.40347878093517959071866248671, −9.765317109662305025008733753625, −8.737308994627787116152159521654, −7.67420944264984030435218778174, −6.62348533578647785006650429958, −5.59891512360326015420997079970, −3.37753736023914227229307692146, −2.05142602988182140675460143922,
1.83106230462030958973673773162, 2.63127562502769833746946957865, 4.81229036110381895036464348372, 6.50562563453023988207755630397, 7.47783753114234095907058017337, 8.934177212558029633442964635946, 9.342903148558019715787723822848, 9.785981707686502650828404551233, 11.50201769605241745786412200857, 12.66880764870647385449248814191