| 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 | \( 1 + (-2.94 + 3.39i)T + (-13.8 - 96.0i)T^{2} \) |
| 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
−11.27303213151653967584959004934, −9.951328813530417698897771168092, −9.318696743037096056007897030234, −8.778254430718512896163541465583, −7.22901407982240908859117824606, −6.57030384767426697065350293258, −4.76507092097691136116804806044, −3.67234468000184776769107818446, −2.84546334762979019032172939554, −1.69699755601969348475178678794,
1.00323932215263171528528352413, 3.28148395141047646452073271398, 4.43224168336977479260744688466, 5.51405079334501780367169412667, 6.75819506629554416401012947628, 7.45251831976650922315775752528, 8.047091727263774493451249631511, 8.953585563858691166013090214638, 9.698101604810122288153342094874, 11.08663701768767497786894090478
