L(s) = 1 | + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (0.623 + 1.07i)5-s + (−0.5 + 0.866i)6-s + (−2.27 − 1.34i)7-s − 0.999·8-s + 9-s + 1.24·10-s − 2.49·11-s + (0.499 + 0.866i)12-s + (−0.785 − 3.51i)13-s + (−2.30 + 1.30i)14-s + (−0.623 − 1.07i)15-s + (−0.5 + 0.866i)16-s + (−0.247 − 0.428i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (0.278 + 0.482i)5-s + (−0.204 + 0.353i)6-s + (−0.861 − 0.507i)7-s − 0.353·8-s + 0.333·9-s + 0.394·10-s − 0.752·11-s + (0.144 + 0.249i)12-s + (−0.217 − 0.976i)13-s + (−0.615 + 0.348i)14-s + (−0.160 − 0.278i)15-s + (−0.125 + 0.216i)16-s + (−0.0600 − 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0257968 + 0.476417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0257968 + 0.476417i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + (2.27 + 1.34i)T \) |
| 13 | \( 1 + (0.785 + 3.51i)T \) |
good | 5 | \( 1 + (-0.623 - 1.07i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 17 | \( 1 + (0.247 + 0.428i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 7.67T + 19T^{2} \) |
| 23 | \( 1 + (0.224 - 0.388i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.71 + 6.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.06 - 7.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.37 - 2.37i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.87 + 3.24i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.47 + 2.55i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.29 - 5.71i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.86 + 6.69i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.727 - 1.26i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 - 0.0276T + 67T^{2} \) |
| 71 | \( 1 + (-4.68 + 8.12i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.07 - 8.78i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.93 + 8.54i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 + (-5.74 + 9.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.509 - 0.883i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.45073831686653567235133032318, −9.984922913747272696423466246436, −8.732667877719737847906255102475, −7.46900278405473406711140545850, −6.47323740074298347061270520110, −5.71493359495347451579022329100, −4.58061126985932387520000782322, −3.41859210599648905175481375319, −2.30036888111904558541440380672, −0.24349707769701556348661131074,
2.22507047565512331569853501570, 3.83606540649433245123022017920, 4.92063839635828558199678021956, 5.78114836600990286504409952021, 6.54213684573096092886604001218, 7.43699028913705864130689625417, 8.738457051728435427201872528083, 9.281591487409931205493001606064, 10.37890001134887257441892537741, 11.31234882714553528367836798478