| L(s) = 1 | + (−2.18 + 1.12i)2-s + (−0.786 + 0.618i)3-s + (2.35 − 3.30i)4-s + (0.783 − 0.746i)5-s + (1.02 − 2.23i)6-s + (−0.0385 − 2.64i)7-s + (−0.718 + 4.99i)8-s + (0.235 − 0.971i)9-s + (−0.870 + 2.51i)10-s + (3.13 + 1.61i)11-s + (0.192 + 4.04i)12-s + (−3.33 − 3.85i)13-s + (3.06 + 5.74i)14-s + (−0.153 + 1.07i)15-s + (−1.41 − 4.08i)16-s + (−2.34 + 0.223i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 + 0.797i)2-s + (−0.453 + 0.356i)3-s + (1.17 − 1.65i)4-s + (0.350 − 0.333i)5-s + (0.417 − 0.913i)6-s + (−0.0145 − 0.999i)7-s + (−0.254 + 1.76i)8-s + (0.0785 − 0.323i)9-s + (−0.275 + 0.795i)10-s + (0.945 + 0.487i)11-s + (0.0556 + 1.16i)12-s + (−0.925 − 1.06i)13-s + (0.819 + 1.53i)14-s + (−0.0397 + 0.276i)15-s + (−0.353 − 1.02i)16-s + (−0.567 + 0.0542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.105804 - 0.153188i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105804 - 0.153188i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.786 - 0.618i)T \) |
| 7 | \( 1 + (0.0385 + 2.64i)T \) |
| 23 | \( 1 + (4.78 + 0.249i)T \) |
| good | 2 | \( 1 + (2.18 - 1.12i)T + (1.16 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-0.783 + 0.746i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (-3.13 - 1.61i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (3.33 + 3.85i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.34 - 0.223i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (7.58 + 0.724i)T + (18.6 + 3.59i)T^{2} \) |
| 29 | \( 1 + (4.23 - 9.27i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-4.20 + 1.68i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (-2.35 + 9.72i)T + (-32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (8.43 + 2.47i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.770 - 5.35i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-4.18 + 7.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.43 + 1.04i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (-0.454 + 1.31i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (0.773 + 0.608i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (-0.195 + 4.09i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (1.44 + 0.925i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.181 + 0.255i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-3.13 + 0.604i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (4.53 - 1.33i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-8.48 - 3.39i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (7.28 + 2.14i)T + (81.6 + 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.49630001282281270584182843765, −9.692680408714476557485300505114, −9.036248522759120946808011475436, −8.025457011368439452840605921933, −7.08656982034211876963710331949, −6.44590211488556080014609941220, −5.30439521153558104895130784955, −4.06547140741978392410385558114, −1.75525790332196733821140861968, −0.18842911546940758843506282080,
1.84370997260876204204376961895, 2.52549826880239358167626922535, 4.34027171524891216823344790985, 6.18869083887676964702616161894, 6.66149674025280002651753014949, 8.031832065219435119236858346473, 8.712487670579415637515499152381, 9.568042574085350139040090523264, 10.22104918089161740400265330593, 11.28443019551196952673890554891
