| L(s) = 1 | + (−0.261 + 1.81i)2-s + (0.909 − 0.415i)3-s + (−1.32 − 0.388i)4-s + (−1.39 − 1.61i)5-s + (0.517 + 1.76i)6-s + (0.194 − 2.63i)7-s + (−0.473 + 1.03i)8-s + (0.654 − 0.755i)9-s + (3.30 − 2.12i)10-s + (5.55 − 0.799i)11-s + (−1.36 + 0.196i)12-s + (−2.77 − 4.32i)13-s + (4.75 + 1.04i)14-s + (−1.94 − 0.886i)15-s + (−4.08 − 2.62i)16-s + (−0.617 + 0.181i)17-s + ⋯ |
| L(s) = 1 | + (−0.185 + 1.28i)2-s + (0.525 − 0.239i)3-s + (−0.662 − 0.194i)4-s + (−0.625 − 0.721i)5-s + (0.211 + 0.720i)6-s + (0.0734 − 0.997i)7-s + (−0.167 + 0.366i)8-s + (0.218 − 0.251i)9-s + (1.04 − 0.670i)10-s + (1.67 − 0.240i)11-s + (−0.394 + 0.0566i)12-s + (−0.770 − 1.19i)13-s + (1.26 + 0.279i)14-s + (−0.501 − 0.228i)15-s + (−1.02 − 0.656i)16-s + (−0.149 + 0.0439i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44507 + 0.224893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44507 + 0.224893i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.194 + 2.63i)T \) |
| 23 | \( 1 + (-4.55 - 1.49i)T \) |
| good | 2 | \( 1 + (0.261 - 1.81i)T + (-1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (1.39 + 1.61i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-5.55 + 0.799i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.77 + 4.32i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.617 - 0.181i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-5.98 - 1.75i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (2.25 - 0.663i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.95 - 1.35i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.96 + 4.30i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.12 + 3.57i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (2.53 - 1.15i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 7.97iT - 47T^{2} \) |
| 53 | \( 1 + (4.63 - 7.20i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.0452 - 0.0704i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.78 + 6.09i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.832 - 0.119i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.233 + 1.62i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.78 - 9.46i)T + (-61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-0.516 - 0.804i)T + (-32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (7.27 - 8.39i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.29 - 5.03i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.94 - 3.39i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.08663701768767497786894090478, −9.698101604810122288153342094874, −8.953585563858691166013090214638, −8.047091727263774493451249631511, −7.45251831976650922315775752528, −6.75819506629554416401012947628, −5.51405079334501780367169412667, −4.43224168336977479260744688466, −3.28148395141047646452073271398, −1.00323932215263171528528352413,
1.69699755601969348475178678794, 2.84546334762979019032172939554, 3.67234468000184776769107818446, 4.76507092097691136116804806044, 6.57030384767426697065350293258, 7.22901407982240908859117824606, 8.778254430718512896163541465583, 9.318696743037096056007897030234, 9.951328813530417698897771168092, 11.27303213151653967584959004934
