| L(s) = 1 | + (0.617 − 2.54i)2-s + (−0.327 + 0.945i)3-s + (−4.31 − 2.22i)4-s + (3.40 + 1.36i)5-s + (2.20 + 1.41i)6-s + (2.47 + 0.943i)7-s + (−4.89 + 5.64i)8-s + (−0.786 − 0.618i)9-s + (5.57 − 7.82i)10-s + (1.13 + 4.67i)11-s + (3.51 − 3.34i)12-s + (1.90 + 4.18i)13-s + (3.92 − 5.70i)14-s + (−2.40 + 2.77i)15-s + (5.71 + 8.02i)16-s + (0.0881 − 1.85i)17-s + ⋯ |
| L(s) = 1 | + (0.436 − 1.79i)2-s + (−0.188 + 0.545i)3-s + (−2.15 − 1.11i)4-s + (1.52 + 0.609i)5-s + (0.899 + 0.577i)6-s + (0.934 + 0.356i)7-s + (−1.73 + 1.99i)8-s + (−0.262 − 0.206i)9-s + (1.76 − 2.47i)10-s + (0.342 + 1.40i)11-s + (1.01 − 0.967i)12-s + (0.529 + 1.15i)13-s + (1.04 − 1.52i)14-s + (−0.620 + 0.715i)15-s + (1.42 + 2.00i)16-s + (0.0213 − 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63694 - 1.14208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63694 - 1.14208i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.327 - 0.945i)T \) |
| 7 | \( 1 + (-2.47 - 0.943i)T \) |
| 23 | \( 1 + (3.00 + 3.73i)T \) |
| good | 2 | \( 1 + (-0.617 + 2.54i)T + (-1.77 - 0.916i)T^{2} \) |
| 5 | \( 1 + (-3.40 - 1.36i)T + (3.61 + 3.45i)T^{2} \) |
| 11 | \( 1 + (-1.13 - 4.67i)T + (-9.77 + 5.04i)T^{2} \) |
| 13 | \( 1 + (-1.90 - 4.18i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.0881 + 1.85i)T + (-16.9 - 1.61i)T^{2} \) |
| 19 | \( 1 + (0.159 + 3.34i)T + (-18.9 + 1.80i)T^{2} \) |
| 29 | \( 1 + (2.28 + 1.46i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (8.29 + 1.59i)T + (28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (2.06 + 1.62i)T + (8.72 + 35.9i)T^{2} \) |
| 41 | \( 1 + (1.03 + 7.21i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.60 + 3.01i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-0.722 + 1.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.56 + 0.531i)T + (52.0 - 10.0i)T^{2} \) |
| 59 | \( 1 + (-3.77 + 5.30i)T + (-19.2 - 55.7i)T^{2} \) |
| 61 | \( 1 + (0.852 + 2.46i)T + (-47.9 + 37.7i)T^{2} \) |
| 67 | \( 1 + (4.44 + 4.23i)T + (3.18 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-5.75 + 1.69i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 5.71i)T + (42.3 + 59.4i)T^{2} \) |
| 79 | \( 1 + (7.42 + 0.709i)T + (77.5 + 14.9i)T^{2} \) |
| 83 | \( 1 + (-0.941 + 6.54i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (11.9 - 2.30i)T + (82.6 - 33.0i)T^{2} \) |
| 97 | \( 1 + (-1.04 - 7.30i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.85099793620723103300266996738, −10.14016342572016577780887218309, −9.373437008019037510937921880307, −8.934646775044909479074359101318, −6.88453041810049852171159368355, −5.60604088358488704359304256542, −4.82204880817320212561950378595, −3.89033507605037735533301568044, −2.24442543731383493922600141992, −1.88624747174433132769217652400,
1.33986256473618343481635334681, 3.66494030007451484295989038768, 5.16529710079795222847650912826, 5.74048052871149710266371321200, 6.17161219023626209481235958917, 7.45483601966680606099496929955, 8.337524255919363115840021103956, 8.774038183673420394947289660180, 10.01246878703374285084728298480, 11.16941486698476819547500499807
