| L(s) = 1 | + (−0.386 − 2.69i)2-s + (−0.909 − 0.415i)3-s + (−5.16 + 1.51i)4-s + (0.820 − 0.946i)5-s + (−0.765 + 2.60i)6-s + (−0.989 + 2.45i)7-s + (3.82 + 8.37i)8-s + (0.654 + 0.755i)9-s + (−2.86 − 1.84i)10-s + (3.33 + 0.479i)11-s + (5.33 + 0.766i)12-s + (−3.47 + 5.40i)13-s + (6.98 + 1.71i)14-s + (−1.13 + 0.520i)15-s + (11.9 − 7.70i)16-s + (0.0646 + 0.0189i)17-s + ⋯ |
| L(s) = 1 | + (−0.273 − 1.90i)2-s + (−0.525 − 0.239i)3-s + (−2.58 + 0.758i)4-s + (0.366 − 0.423i)5-s + (−0.312 + 1.06i)6-s + (−0.373 + 0.927i)7-s + (1.35 + 2.96i)8-s + (0.218 + 0.251i)9-s + (−0.905 − 0.582i)10-s + (1.00 + 0.144i)11-s + (1.53 + 0.221i)12-s + (−0.962 + 1.49i)13-s + (1.86 + 0.457i)14-s + (−0.294 + 0.134i)15-s + (2.99 − 1.92i)16-s + (0.0156 + 0.00460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620591 - 0.212262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620591 - 0.212262i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.989 - 2.45i)T \) |
| 23 | \( 1 + (4.37 - 1.97i)T \) |
| good | 2 | \( 1 + (0.386 + 2.69i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.820 + 0.946i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.33 - 0.479i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.47 - 5.40i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.0646 - 0.0189i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.91 + 0.563i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (1.67 + 0.492i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (5.40 - 2.46i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.98 + 6.05i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.621 + 0.538i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.52 - 1.60i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 12.4iT - 47T^{2} \) |
| 53 | \( 1 + (-1.75 - 2.73i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.79 + 2.78i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.80 + 3.95i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (6.30 - 0.906i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.25 - 8.69i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.91 - 13.3i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (3.74 - 5.83i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.47 - 2.86i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-6.18 + 13.5i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 12.4i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.33514168344848021658921788199, −9.872547228419864729117146624057, −9.363433018156794880038529817349, −8.912109475285012891670465705343, −7.42090553572693923459109074966, −5.92357366729365096812374977562, −4.84256737149759899893151188496, −3.86903236367760148625571722302, −2.38424688000195145707413506142, −1.48287569162741718973272233539,
0.50625334324298941717808444063, 3.68813921612493237804785920473, 4.75806052816202809492892648519, 5.79798125536807887527299122607, 6.44953436284106954064828708733, 7.30493308266361103852913523917, 8.002877868374455855370738349049, 9.255720760139652818808926445388, 10.02213486346304141835222470456, 10.50454951186785386354799556476
