| L(s) = 1 | + (−1.30 − 0.951i)2-s + (−2.95 − 1.31i)3-s + (0.190 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (2.61 + 4.53i)6-s + (0.157 − 0.175i)7-s + (−0.690 + 2.12i)8-s + (4.99 + 5.55i)9-s + (1.47 − 0.658i)10-s + (−1.95 − 0.415i)11-s + (0.209 − 1.98i)12-s + (0.338 + 3.21i)13-s + (−0.373 + 0.0794i)14-s + (2.61 − 1.90i)15-s + (3.92 − 2.85i)16-s + (−0.747 + 0.158i)17-s + ⋯ |
| L(s) = 1 | + (−0.925 − 0.672i)2-s + (−1.70 − 0.759i)3-s + (0.0954 + 0.293i)4-s + (−0.223 + 0.387i)5-s + (1.06 + 1.85i)6-s + (0.0597 − 0.0663i)7-s + (−0.244 + 0.751i)8-s + (1.66 + 1.85i)9-s + (0.467 − 0.208i)10-s + (−0.589 − 0.125i)11-s + (0.0603 − 0.574i)12-s + (0.0938 + 0.892i)13-s + (−0.0998 + 0.0212i)14-s + (0.675 − 0.491i)15-s + (0.981 − 0.713i)16-s + (−0.181 + 0.0385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000462493 + 0.208206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000462493 + 0.208206i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (1.30 + 0.951i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (2.95 + 1.31i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.157 + 0.175i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (1.95 + 0.415i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.338 - 3.21i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (0.747 - 0.158i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.233 + 2.22i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.76 + 5.42i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.23 + 1.62i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.39 + 2.84i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.129 - 1.22i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (2 - 1.45i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (7.00 + 7.78i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-2.04 - 0.909i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 8.18T + 61T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.14 + 6.82i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (8.28 + 1.76i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-11.4 + 2.43i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (13.6 - 6.07i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.61 - 11.1i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (4.92 + 15.1i)T + (-78.4 + 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.913737553296989845008047448902, −8.912476505579886803445758920083, −7.84733721006105728717279447341, −7.05574185086875396355502102351, −6.27260602718360331329765466662, −5.36238127818924991538182404450, −4.49684487333086695161552711886, −2.58292996710064620528850832071, −1.43357197271161183365410659094, −0.23567692889004931292023260099,
0.938791577748437918209742390881, 3.49471874149431193227568583481, 4.55066861970383148489962242127, 5.46652886372924600501066242669, 6.08700178299702460366128814983, 7.10492888017794513395171620279, 7.85648123916705843052285899481, 8.860907903832274713085315712290, 9.720557659277721424725403276313, 10.30108373006669157371349266687
