L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (1.97 + 1.05i)5-s − 5.04·7-s + (0.309 − 0.951i)8-s + (−0.974 − 2.01i)10-s + (4.35 + 3.16i)11-s + (2.78 − 2.02i)13-s + (4.08 + 2.96i)14-s + (−0.809 + 0.587i)16-s + (−1.15 + 3.54i)17-s + (−1.51 + 4.65i)19-s + (−0.394 + 2.20i)20-s + (−1.66 − 5.11i)22-s + (1.82 + 1.32i)23-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.881 + 0.472i)5-s − 1.90·7-s + (0.109 − 0.336i)8-s + (−0.308 − 0.636i)10-s + (1.31 + 0.953i)11-s + (0.771 − 0.560i)13-s + (1.09 + 0.793i)14-s + (−0.202 + 0.146i)16-s + (−0.279 + 0.860i)17-s + (−0.346 + 1.06i)19-s + (−0.0882 + 0.492i)20-s + (−0.354 − 1.09i)22-s + (0.381 + 0.276i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.932122 + 0.381063i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.932122 + 0.381063i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.97 - 1.05i)T \) |
good | 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 + (-4.35 - 3.16i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.78 + 2.02i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.15 - 3.54i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.51 - 4.65i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.82 - 1.32i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.153 + 0.473i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.05 - 6.33i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.558 - 0.405i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-9.12 + 6.62i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.179T + 43T^{2} \) |
| 47 | \( 1 + (-3.34 - 10.3i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.526 - 1.61i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (4.45 - 3.24i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.55 + 6.21i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.86 + 8.81i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.15 + 3.55i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.92 + 2.85i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.52 + 7.77i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.72 + 11.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.89 - 1.37i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.584 - 1.79i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.76773218682646486685818564706, −10.31291486958019822184679782350, −9.360801485749272146653957072324, −9.027723258594577417503126375378, −7.42838660982013797177431184420, −6.42983283498474141480546310084, −6.02473422829832469774231328023, −3.93703335182082531415813143998, −3.07608569500526404908810815007, −1.61456464674870836602817620325,
0.805152250360772865981541688996, 2.69210678649848187469455842909, 4.08242699775367297454104590181, 5.71376464608132463498095101773, 6.42382269360311494846126134394, 6.93079606222077049499160840182, 8.705764501156905704280242835984, 9.215609909882001019883365913901, 9.642578199401690288740057546444, 10.82314035601531213597274725053