L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.280 − 1.70i)3-s + (−0.499 + 0.866i)4-s + (−0.0759 + 0.0438i)5-s + (1.33 − 1.09i)6-s + (−2.62 − 0.361i)7-s − 0.999·8-s + (−2.84 + 0.960i)9-s + (−0.0759 − 0.0438i)10-s + (−2.83 + 4.90i)11-s + (1.62 + 0.611i)12-s + (−2.43 + 2.66i)13-s + (−0.997 − 2.45i)14-s + (0.0962 + 0.117i)15-s + (−0.5 − 0.866i)16-s + (1.33 − 2.31i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.162 − 0.986i)3-s + (−0.249 + 0.433i)4-s + (−0.0339 + 0.0196i)5-s + (0.546 − 0.448i)6-s + (−0.990 − 0.136i)7-s − 0.353·8-s + (−0.947 + 0.320i)9-s + (−0.0240 − 0.0138i)10-s + (−0.853 + 1.47i)11-s + (0.467 + 0.176i)12-s + (−0.674 + 0.738i)13-s + (−0.266 − 0.654i)14-s + (0.0248 + 0.0303i)15-s + (−0.125 − 0.216i)16-s + (0.323 − 0.561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.163162 + 0.538982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.163162 + 0.538982i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.280 + 1.70i)T \) |
| 7 | \( 1 + (2.62 + 0.361i)T \) |
| 13 | \( 1 + (2.43 - 2.66i)T \) |
good | 5 | \( 1 + (0.0759 - 0.0438i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.83 - 4.90i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.33 + 2.31i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.32 - 5.76i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.06 + 0.613i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.84iT - 29T^{2} \) |
| 31 | \( 1 + (0.441 - 0.765i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.456 + 0.263i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.99iT - 41T^{2} \) |
| 43 | \( 1 + 9.69T + 43T^{2} \) |
| 47 | \( 1 + (7.45 - 4.30i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (12.1 + 7.00i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.11 - 1.79i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.01 + 1.16i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.54 + 3.77i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 + (4.11 - 7.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.19 - 12.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.86iT - 83T^{2} \) |
| 89 | \( 1 + (-2.93 + 1.69i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−11.50959636494262698200947060071, −9.995343377899525789890015131385, −9.507746534708429519672197975124, −8.063510030915383450256834366599, −7.35255426633282817408665726413, −6.81925576688133768307116011461, −5.75066044227063869588270180370, −4.84490313410796027001079327327, −3.37404340437769342303435130212, −2.06721037793951850364324419730,
0.27629295839382699343345871123, 3.01554700312171982135347345502, 3.23054747020859620618774905829, 4.78622475731496407442986695874, 5.54211099724495126557377426396, 6.42014667570542235694750258467, 8.016081872445648228569479627097, 8.959611964945410920327164652285, 9.821875926252801627974762817755, 10.44374469404153285068269469342