| L(s) = 1 | + (0.366 + 1.36i)2-s + (−1.33 + 1.10i)3-s + (−2 − 1.41i)6-s + (−2.38 − 1.15i)7-s + (2 + 1.99i)8-s + (0.548 − 2.94i)9-s + (−3.67 + 2.12i)11-s + (−3.53 + 3.53i)13-s + (0.707 − 3.67i)14-s + (−1.99 + 3.46i)16-s + (−1.36 − 0.366i)17-s + (4.22 − 0.330i)18-s + (−6.06 − 3.5i)19-s + (4.44 − 1.09i)21-s + (−4.24 − 4.24i)22-s + (−4.09 + 1.09i)23-s + ⋯ |
| L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.769 + 0.639i)3-s + (−0.816 − 0.577i)6-s + (−0.899 − 0.436i)7-s + (0.707 + 0.707i)8-s + (0.182 − 0.983i)9-s + (−1.10 + 0.639i)11-s + (−0.980 + 0.980i)13-s + (0.188 − 0.981i)14-s + (−0.499 + 0.866i)16-s + (−0.331 − 0.0887i)17-s + (0.996 − 0.0779i)18-s + (−1.39 − 0.802i)19-s + (0.970 − 0.239i)21-s + (−0.904 − 0.904i)22-s + (−0.854 + 0.228i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.187308 - 0.443897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187308 - 0.443897i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.33 - 1.10i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.38 + 1.15i)T \) |
| good | 2 | \( 1 + (-0.366 - 1.36i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.53 - 3.53i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.36 + 0.366i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (6.06 + 3.5i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.09 - 1.09i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.965 + 0.258i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-3.53 + 3.53i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.46 + 5.46i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.732 - 2.73i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.258 + 0.965i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (6.76 + 1.81i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.06 + 3.5i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1 + i)T + 83iT^{2} \) |
| 89 | \( 1 + (7.77 - 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.07 + 7.07i)T + 97iT^{2} \) |
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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
−11.29043587934375352316777425237, −10.35914276163888547727296018008, −9.912816697855094916142647161897, −8.716850564328786674855091676807, −7.38246672697746239466606492674, −6.74079722012130617124155524448, −6.03633291097996436179278862421, −4.84190231806346028365635630783, −4.33128829542689080849937271635, −2.46024004159058986217061371195,
0.25663987346834354596711321596, 2.19965650760220021533538898237, 2.97993613209101044209735503330, 4.47546784922258891045712145105, 5.69542535969848004527730074195, 6.46486417124305658041614111876, 7.57170072134059327418493794182, 8.381259547371103781352179970176, 10.07947039855675152551327213994, 10.36827790558860194335780887657
