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
−9.657756893702986201759218320771, −8.491677782232552694231415706065, −8.027382386670021601496736507693, −7.05450527446159565153980669696, −6.04584554760532082087650235369, −5.43240275147190804650305990858, −4.45505166846529717027683983751, −3.66516411479278706742311765831, −2.20705405902293397210486293381, −1.33307648517861311081827693162,
0.62341083741715541956043776308, 2.52025647958306487466751152811, 2.79553370923834536353293614416, 4.29648927015540688758092502634, 5.04966969264627357157604158314, 6.10976962063568839156611912040, 6.82404610126947459825214127948, 7.39554198926179615503978294676, 8.509782118843760520007398093869, 9.193713381170703994465212294195