L(s) = 1 | + (1.13 + 0.847i)2-s + (1.71 + 0.242i)3-s + (0.562 + 1.91i)4-s + (1.73 + 1.72i)6-s − 3.08·7-s + (−0.990 + 2.64i)8-s + (2.88 + 0.831i)9-s + 2.54i·11-s + (0.499 + 3.42i)12-s + 5.06·13-s + (−3.49 − 2.61i)14-s + (−3.36 + 2.15i)16-s + 0.214·17-s + (2.55 + 3.38i)18-s − 2.60·19-s + ⋯ |
L(s) = 1 | + (0.800 + 0.599i)2-s + (0.990 + 0.139i)3-s + (0.281 + 0.959i)4-s + (0.708 + 0.705i)6-s − 1.16·7-s + (−0.350 + 0.936i)8-s + (0.960 + 0.277i)9-s + 0.767i·11-s + (0.144 + 0.989i)12-s + 1.40·13-s + (−0.934 − 0.700i)14-s + (−0.841 + 0.539i)16-s + 0.0519·17-s + (0.602 + 0.797i)18-s − 0.598·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0345 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0345 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08953 + 2.01850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08953 + 2.01850i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.13 - 0.847i)T \) |
| 3 | \( 1 + (-1.71 - 0.242i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 - 0.214T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 - 4.58iT - 31T^{2} \) |
| 37 | \( 1 + 7.67T + 37T^{2} \) |
| 41 | \( 1 + 9.26iT - 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 9.51iT - 53T^{2} \) |
| 59 | \( 1 - 0.428iT - 59T^{2} \) |
| 61 | \( 1 - 1.11iT - 61T^{2} \) |
| 67 | \( 1 - 2.35iT - 67T^{2} \) |
| 71 | \( 1 - 6.12T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 12.4iT - 89T^{2} \) |
| 97 | \( 1 + 8.04iT - 97T^{2} \) |
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
−10.71356283141388794192697939410, −9.988563767555620829727020121887, −8.648186233134838333661349342535, −8.501138200339382520960510839569, −6.90167997747146979937006839818, −6.72752413115503427881439354829, −5.31923740396427176342322978338, −4.06405318266603869764486192166, −3.43431640202337890739336404084, −2.25988201301460327798960010772,
1.30468648264045463704467631871, 2.91558887274560489120890592849, 3.43734152495663867763391534559, 4.47232437547612972877834228686, 6.11391298780142968834314677453, 6.46057921790218864989501724113, 7.87097958992238139228596909543, 8.919562130783934753777206271484, 9.612527464759220068984202741697, 10.48407674520882727061870924168