| L(s) = 1 | + (0.733 − 1.02i)2-s + (0.235 − 0.971i)3-s + (0.131 + 0.380i)4-s + (0.00357 − 0.0751i)5-s + (−0.827 − 0.955i)6-s + (1.84 + 1.89i)7-s + (2.91 + 0.855i)8-s + (−0.888 − 0.458i)9-s + (−0.0747 − 0.0587i)10-s + (3.67 + 5.16i)11-s + (0.400 − 0.0382i)12-s + (0.444 − 3.09i)13-s + (3.30 − 0.515i)14-s + (−0.0721 − 0.0211i)15-s + (2.38 − 1.87i)16-s + (−6.91 + 1.33i)17-s + ⋯ |
| L(s) = 1 | + (0.518 − 0.727i)2-s + (0.136 − 0.561i)3-s + (0.0658 + 0.190i)4-s + (0.00160 − 0.0336i)5-s + (−0.337 − 0.389i)6-s + (0.698 + 0.715i)7-s + (1.03 + 0.302i)8-s + (−0.296 − 0.152i)9-s + (−0.0236 − 0.0185i)10-s + (1.10 + 1.55i)11-s + (0.115 − 0.0110i)12-s + (0.123 − 0.857i)13-s + (0.883 − 0.137i)14-s + (−0.0186 − 0.00547i)15-s + (0.595 − 0.468i)16-s + (−1.67 + 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12749 - 0.735852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12749 - 0.735852i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (-1.84 - 1.89i)T \) |
| 23 | \( 1 + (1.04 + 4.68i)T \) |
| good | 2 | \( 1 + (-0.733 + 1.02i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (-0.00357 + 0.0751i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (-3.67 - 5.16i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.444 + 3.09i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (6.91 - 1.33i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (0.610 + 0.117i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (6.63 + 7.66i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-3.91 + 3.72i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.313 - 0.161i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (6.85 + 4.40i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (4.84 - 1.42i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-2.30 - 3.99i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.37 - 0.552i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (-3.25 - 2.55i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (1.30 + 5.38i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (0.716 + 0.0684i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (0.989 + 2.16i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.562 + 1.62i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (5.66 - 2.26i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (1.80 - 1.16i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (7.20 + 6.87i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (-7.31 - 4.69i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.27024881341268073842972205489, −10.21978158279555890262651841464, −9.001635413848480299891796788032, −8.205636044533517595198708267158, −7.24697069798945051818319278348, −6.29362679096383542772059473462, −4.82209566505246877730518387896, −4.08873390626961926528294339428, −2.50789190605670111778641877841, −1.80403790610015015687779180852,
1.49232433347687431202138120490, 3.58650140598994599559538402673, 4.45164875836200391779502313190, 5.34216260668767397931412216889, 6.52822552073175186897658100100, 7.06145972648969416495663545033, 8.483703246644538775949148050160, 9.074485513660877728790162367811, 10.34009470834155658000410383114, 11.18324914545463948256116564655
