| 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
−11.16941486698476819547500499807, −10.01246878703374285084728298480, −8.774038183673420394947289660180, −8.337524255919363115840021103956, −7.45483601966680606099496929955, −6.17161219023626209481235958917, −5.74048052871149710266371321200, −5.16529710079795222847650912826, −3.66494030007451484295989038768, −1.33986256473618343481635334681,
1.88624747174433132769217652400, 2.24442543731383493922600141992, 3.89033507605037735533301568044, 4.82204880817320212561950378595, 5.60604088358488704359304256542, 6.88453041810049852171159368355, 8.934646775044909479074359101318, 9.373437008019037510937921880307, 10.14016342572016577780887218309, 10.85099793620723103300266996738
