L(s) = 1 | + (−0.866 − 0.5i)2-s + (1.62 + 0.608i)3-s + (0.499 + 0.866i)4-s + (−1.94 − 3.36i)5-s + (−1.10 − 1.33i)6-s + (2.09 − 1.60i)7-s − 0.999i·8-s + (2.26 + 1.97i)9-s + 3.89i·10-s + (3.41 + 1.97i)11-s + (0.284 + 1.70i)12-s + (−2.46 + 1.42i)13-s + (−2.62 + 0.343i)14-s + (−1.10 − 6.64i)15-s + (−0.5 + 0.866i)16-s − 0.742·17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.936 + 0.351i)3-s + (0.249 + 0.433i)4-s + (−0.870 − 1.50i)5-s + (−0.449 − 0.546i)6-s + (0.793 − 0.608i)7-s − 0.353i·8-s + (0.753 + 0.657i)9-s + 1.23i·10-s + (1.03 + 0.594i)11-s + (0.0820 + 0.493i)12-s + (−0.684 + 0.395i)13-s + (−0.701 + 0.0919i)14-s + (−0.285 − 1.71i)15-s + (−0.125 + 0.216i)16-s − 0.179·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.949782 - 0.380215i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.949782 - 0.380215i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-1.62 - 0.608i)T \) |
| 7 | \( 1 + (-2.09 + 1.60i)T \) |
good | 5 | \( 1 + (1.94 + 3.36i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.41 - 1.97i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.46 - 1.42i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.742T + 17T^{2} \) |
| 19 | \( 1 + 1.78iT - 19T^{2} \) |
| 23 | \( 1 + (5.41 - 3.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.50 + 1.44i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.04 - 1.75i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.00T + 37T^{2} \) |
| 41 | \( 1 + (-5.24 - 9.08i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.471 + 0.816i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.09 - 1.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (0.0105 + 0.0183i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.13 - 1.23i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.72 + 11.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.94iT - 71T^{2} \) |
| 73 | \( 1 - 4.85iT - 73T^{2} \) |
| 79 | \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.02 + 6.98i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + (16.2 + 9.40i)T + (48.5 + 84.0i)T^{2} \) |
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
−13.12756512040172887931615263406, −12.12037376030140920778037297357, −11.27937732185803586454744386805, −9.739593061322474448266270359184, −9.066950412843359688014421538420, −8.070527265685172916323998125139, −7.36927025757613417420469724718, −4.66426106015470652096624380482, −3.98768681714196133660270242315, −1.62019668701832695227740765842,
2.37252111643370981445405582503, 3.85155398602866355139392476605, 6.18043132116941745820290339410, 7.31272016212396169951795407072, 8.013113512560948855018556327236, 9.030311391300753122748796099716, 10.32538680004834245497628924998, 11.39072086517669874310171684431, 12.24540398019516900574044445913, 14.08457116060493978017432847451