L(s) = 1 | + 1.32·2-s + (0.5 − 0.866i)3-s − 0.245·4-s + (−2.05 + 3.55i)5-s + (0.662 − 1.14i)6-s − 3.69·7-s − 2.97·8-s + (−0.499 − 0.866i)9-s + (−2.71 + 4.71i)10-s + (0.214 − 0.371i)11-s + (−0.122 + 0.212i)12-s + (1.17 + 2.03i)13-s − 4.89·14-s + (2.05 + 3.55i)15-s − 3.44·16-s + (0.765 − 1.32i)17-s + ⋯ |
L(s) = 1 | + 0.936·2-s + (0.288 − 0.499i)3-s − 0.122·4-s + (−0.918 + 1.59i)5-s + (0.270 − 0.468i)6-s − 1.39·7-s − 1.05·8-s + (−0.166 − 0.288i)9-s + (−0.859 + 1.48i)10-s + (0.0647 − 0.112i)11-s + (−0.0354 + 0.0614i)12-s + (0.326 + 0.565i)13-s − 1.30·14-s + (0.530 + 0.918i)15-s − 0.862·16-s + (0.185 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.274481 + 0.712145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.274481 + 0.712145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 157 | \( 1 + (-1.42 - 12.4i)T \) |
good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 + (2.05 - 3.55i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + (-0.214 + 0.371i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.17 - 2.03i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.765 + 1.32i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.35 - 5.81i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 - 6.08T + 29T^{2} \) |
| 31 | \( 1 + (2.68 + 4.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.23 - 7.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.93T + 41T^{2} \) |
| 43 | \( 1 + (-4.73 - 8.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.29 + 5.71i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.67 - 8.09i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 3.06T + 59T^{2} \) |
| 61 | \( 1 + (4.87 - 8.44i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + (-0.536 - 0.929i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.85 + 4.94i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 7.18T + 79T^{2} \) |
| 83 | \( 1 + (5.58 + 9.68i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.330 + 0.572i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.13 - 1.96i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.75044899727587449528938807925, −10.54910379344334818871919858944, −9.694779596924722590709660545315, −8.545998378290338020044999969410, −7.46973119013654617135751450753, −6.41631092760406598072605402211, −6.19652975891274987674637618557, −4.25261947806224326494820026879, −3.39971459950271567504698791453, −2.80147817593918892837119512835,
0.32528831131428822860686810638, 3.08858334637976567047207306610, 3.92758419150715157141587049809, 4.72934151954641026841617560306, 5.55746721698933933843950543070, 6.78336987056804644659082390905, 8.264905198059487914680435718984, 8.933323609719900634069599374898, 9.495828233561386694570403426095, 10.75241681819003043864615240504