L(s) = 1 | − 1.73·3-s − 3.46i·7-s + 2.99·9-s − 3.46·11-s − 4·13-s + 6i·17-s − 3.46i·19-s + 5.99i·21-s + 3.46·23-s − 5.19·27-s − 6i·29-s + 3.46i·31-s + 5.99·33-s + 4·37-s + 6.92·39-s + ⋯ |
L(s) = 1 | − 1.00·3-s − 1.30i·7-s + 0.999·9-s − 1.04·11-s − 1.10·13-s + 1.45i·17-s − 0.794i·19-s + 1.30i·21-s + 0.722·23-s − 1.00·27-s − 1.11i·29-s + 0.622i·31-s + 1.04·33-s + 0.657·37-s + 1.10·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2812116991\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2812116991\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 12iT - 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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
−10.17961646060576290664693340363, −9.550851924248552049879894815737, −8.093903893531448088581522837653, −7.48471459619157969804729289078, −6.71866854605241423728366863602, −5.85155468172535609992734301066, −4.75222893608096037942303067435, −4.30818213461729069166455355256, −2.84981593718342395936760095328, −1.24949974391807442269895498300,
0.14495271801349587772582375727, 2.05035438238152100739931983970, 3.04854332335338325993964436251, 4.68128785864173441191328753785, 5.29820230688059406651349201561, 5.84149255384127535926568052636, 7.05813651340714620350130199679, 7.60142384595202440677427650662, 8.804330476792746120025822901236, 9.559154992475710805620485588468