L(s) = 1 | + (1 + 1.73i)5-s + (1 − 1.73i)11-s − 4·13-s + (3 − 5.19i)17-s + (−4 − 6.92i)19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + 10·29-s + (−2 + 3.46i)31-s + (−3 − 5.19i)37-s + 6·41-s + 4·43-s + (4 + 6.92i)47-s + (1 − 1.73i)53-s + 3.99·55-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (0.301 − 0.522i)11-s − 1.10·13-s + (0.727 − 1.26i)17-s + (−0.917 − 1.58i)19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + 1.85·29-s + (−0.359 + 0.622i)31-s + (−0.493 − 0.854i)37-s + 0.937·41-s + 0.609·43-s + (0.583 + 1.01i)47-s + (0.137 − 0.237i)53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.596894815\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.596894815\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−9.193713381170703994465212294195, −8.509782118843760520007398093869, −7.39554198926179615503978294676, −6.82404610126947459825214127948, −6.10976962063568839156611912040, −5.04966969264627357157604158314, −4.29648927015540688758092502634, −2.79553370923834536353293614416, −2.52025647958306487466751152811, −0.62341083741715541956043776308,
1.33307648517861311081827693162, 2.20705405902293397210486293381, 3.66516411479278706742311765831, 4.45505166846529717027683983751, 5.43240275147190804650305990858, 6.04584554760532082087650235369, 7.05450527446159565153980669696, 8.027382386670021601496736507693, 8.491677782232552694231415706065, 9.657756893702986201759218320771