| L(s) = 1 | + (−0.604 + 1.27i)2-s + (0.153 − 1.72i)3-s + (−1.26 − 1.54i)4-s + (0.0845 − 0.110i)5-s + (2.11 + 1.23i)6-s + (0.741 + 2.76i)7-s + (2.74 − 0.689i)8-s + (−2.95 − 0.529i)9-s + (0.0898 + 0.174i)10-s + (0.545 − 4.14i)11-s + (−2.86 + 1.95i)12-s + (5.29 − 0.696i)13-s + (−3.98 − 0.724i)14-s + (−0.177 − 0.162i)15-s + (−0.775 + 3.92i)16-s − 4.40i·17-s + ⋯ |
| L(s) = 1 | + (−0.427 + 0.904i)2-s + (0.0885 − 0.996i)3-s + (−0.634 − 0.772i)4-s + (0.0378 − 0.0492i)5-s + (0.862 + 0.505i)6-s + (0.280 + 1.04i)7-s + (0.969 − 0.243i)8-s + (−0.984 − 0.176i)9-s + (0.0284 + 0.0552i)10-s + (0.164 − 1.24i)11-s + (−0.825 + 0.563i)12-s + (1.46 − 0.193i)13-s + (−1.06 − 0.193i)14-s + (−0.0457 − 0.0420i)15-s + (−0.193 + 0.981i)16-s − 1.06i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02437 - 0.207301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02437 - 0.207301i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.604 - 1.27i)T \) |
| 3 | \( 1 + (-0.153 + 1.72i)T \) |
| good | 5 | \( 1 + (-0.0845 + 0.110i)T + (-1.29 - 4.82i)T^{2} \) |
| 7 | \( 1 + (-0.741 - 2.76i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.545 + 4.14i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-5.29 + 0.696i)T + (12.5 - 3.36i)T^{2} \) |
| 17 | \( 1 + 4.40iT - 17T^{2} \) |
| 19 | \( 1 + (-2.92 + 7.05i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.24 + 4.66i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (4.43 - 3.40i)T + (7.50 - 28.0i)T^{2} \) |
| 31 | \( 1 + (2.71 + 4.69i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.24 - 5.41i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (2.68 - 10.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.02 - 7.77i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (-2.56 - 1.48i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.32 + 1.37i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 2.65i)T + (-15.2 - 56.9i)T^{2} \) |
| 61 | \( 1 + (1.15 - 0.889i)T + (15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (-0.525 - 3.98i)T + (-64.7 + 17.3i)T^{2} \) |
| 71 | \( 1 + (-0.0104 + 0.0104i)T - 71iT^{2} \) |
| 73 | \( 1 + (5.52 + 5.52i)T + 73iT^{2} \) |
| 79 | \( 1 + (-5.52 - 3.19i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.83 + 7.59i)T + (-21.4 + 80.1i)T^{2} \) |
| 89 | \( 1 + (9.12 - 9.12i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.19 + 3.79i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.44113697812994201314764031071, −11.17565826488116081856961534087, −9.224460201744573065290677558304, −8.795042775031084076092208025192, −7.960798891463053595526521330728, −6.81874718146701906636161213078, −5.97400628788031195814036135209, −5.14883451696685799843020823211, −2.98268239764091611473512575310, −1.04321929849312604970092862249,
1.70046522860876470514386824720, 3.79342310640065154251006109428, 4.00402379722542166989974588556, 5.57469690578442503606436545618, 7.33209930244120307657056543034, 8.329351871375983064464686677258, 9.266006111518497154360062983852, 10.32520037403350714110477004189, 10.55943082023505518514190151391, 11.61851619938990405374728100516
