| L(s) = 1 | + (0.0565 − 0.0444i)2-s + (−0.814 + 0.580i)3-s + (−0.470 + 1.93i)4-s + (4.36 + 0.841i)5-s + (−0.0202 + 0.0689i)6-s + (1.52 + 2.16i)7-s + (0.119 + 0.261i)8-s + (0.327 − 0.945i)9-s + (0.284 − 0.146i)10-s + (−2.94 + 3.74i)11-s + (−0.741 − 1.85i)12-s + (2.81 − 4.37i)13-s + (0.182 + 0.0546i)14-s + (−4.04 + 1.84i)15-s + (−3.52 − 1.81i)16-s + (2.37 − 2.26i)17-s + ⋯ |
| L(s) = 1 | + (0.0399 − 0.0314i)2-s + (−0.470 + 0.334i)3-s + (−0.235 + 0.969i)4-s + (1.95 + 0.376i)5-s + (−0.00826 + 0.0281i)6-s + (0.575 + 0.817i)7-s + (0.0421 + 0.0923i)8-s + (0.109 − 0.315i)9-s + (0.0898 − 0.0463i)10-s + (−0.887 + 1.12i)11-s + (−0.214 − 0.534i)12-s + (0.780 − 1.21i)13-s + (0.0486 + 0.0146i)14-s + (−1.04 + 0.476i)15-s + (−0.881 − 0.454i)16-s + (0.575 − 0.548i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0873 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0873 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22095 + 1.11855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22095 + 1.11855i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 - 0.580i)T \) |
| 7 | \( 1 + (-1.52 - 2.16i)T \) |
| 23 | \( 1 + (3.81 + 2.90i)T \) |
| good | 2 | \( 1 + (-0.0565 + 0.0444i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-4.36 - 0.841i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (2.94 - 3.74i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.81 + 4.37i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.37 + 2.26i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.63 + 2.51i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (1.25 + 0.368i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.457 - 4.78i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (3.22 + 1.11i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (1.13 + 0.986i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.41 - 2.92i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-3.38 + 1.95i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.69 - 0.176i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (2.25 + 4.38i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (0.636 - 0.893i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-1.84 + 4.61i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.79 + 12.4i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.87 - 0.939i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-4.30 - 0.205i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (7.78 + 8.98i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (3.35 - 0.320i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (5.86 - 6.77i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.95052664912508679180520170168, −10.35156074840365851990695671181, −9.473583614139271913235247921895, −8.661159598084352437759425972204, −7.57732529286520106581333024487, −6.37617785046402036277102078624, −5.44203404042247539592617461141, −4.81530393614187920941148745743, −2.99312045784056002705531763507, −2.09408480095035527739846292974,
1.19305602340965974416096549483, 1.99534306751682057079686340744, 4.24880584953234507755506058075, 5.54050889919645040900178551612, 5.82069136977342730663204225105, 6.68803507846992233048157231787, 8.202780079648051867326078677487, 9.139626703549586998178350209434, 10.09579820327338894838085269388, 10.59850919603778863989468698019
