| L(s) = 1 | + (−0.195 − 1.36i)2-s + (0.909 + 0.415i)3-s + (0.105 − 0.0310i)4-s + (1.31 − 1.52i)5-s + (0.387 − 1.31i)6-s + (1.67 + 2.04i)7-s + (−1.20 − 2.63i)8-s + (0.654 + 0.755i)9-s + (−2.33 − 1.49i)10-s + (−5.20 − 0.748i)11-s + (0.109 + 0.0157i)12-s + (2.27 − 3.54i)13-s + (2.46 − 2.67i)14-s + (1.83 − 0.837i)15-s + (−3.16 + 2.03i)16-s + (−0.126 − 0.0370i)17-s + ⋯ |
| L(s) = 1 | + (−0.138 − 0.962i)2-s + (0.525 + 0.239i)3-s + (0.0529 − 0.0155i)4-s + (0.590 − 0.681i)5-s + (0.158 − 0.538i)6-s + (0.632 + 0.774i)7-s + (−0.426 − 0.932i)8-s + (0.218 + 0.251i)9-s + (−0.737 − 0.473i)10-s + (−1.56 − 0.225i)11-s + (0.0315 + 0.00453i)12-s + (0.632 − 0.983i)13-s + (0.657 − 0.715i)14-s + (0.473 − 0.216i)15-s + (−0.792 + 0.509i)16-s + (−0.0306 − 0.00898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000452 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.000452 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35440 - 1.35502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35440 - 1.35502i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (-1.67 - 2.04i)T \) |
| 23 | \( 1 + (-0.0737 - 4.79i)T \) |
| good | 2 | \( 1 + (0.195 + 1.36i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-1.31 + 1.52i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (5.20 + 0.748i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.27 + 3.54i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.126 + 0.0370i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.51 + 2.20i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.12 - 0.329i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (7.25 - 3.31i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.47 + 3.87i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.18 - 1.02i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (5.46 + 2.49i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 1.35iT - 47T^{2} \) |
| 53 | \( 1 + (0.912 + 1.42i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.17 - 3.38i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.75 + 8.22i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.970 - 0.139i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.54 - 10.7i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.68 - 12.5i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (7.46 - 11.6i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.68 - 8.87i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.67 + 16.8i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (11.1 - 12.8i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.84267761236445959639527369622, −9.848730654902142074456817122874, −9.228846340784949618763026975744, −8.290020193059238137900227153325, −7.39288974898889973354028761714, −5.56221979262929057811024035881, −5.24753696347769645417630196036, −3.39739266802536582076993671103, −2.55404747086925248461384564756, −1.32773911235088776536181111583,
1.95720791981342709265861142000, 3.09451829485825445128134456459, 4.73078330209789424241705615364, 5.87192990896233078340172300165, 6.78786550158596555011671874123, 7.61538223414574339860607285599, 8.065452494506480791748586443656, 9.257284493704262189991338355286, 10.32094957395594079382475482805, 11.02839717134235521632045561905
