L(s) = 1 | + (0.5 − 0.866i)2-s + (1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 3.46i·5-s + (1.5 − 0.866i)6-s + (2.5 + 0.866i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−2.99 − 1.73i)10-s + (−1.5 + 2.59i)11-s − 1.73i·12-s + (3.5 − 0.866i)13-s + (2 − 1.73i)14-s + (2.99 − 5.19i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 1.54i·5-s + (0.612 − 0.353i)6-s + (0.944 + 0.327i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.948 − 0.547i)10-s + (−0.452 + 0.783i)11-s − 0.499i·12-s + (0.970 − 0.240i)13-s + (0.534 − 0.462i)14-s + (0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03406 - 1.28275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03406 - 1.28275i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 13 | \( 1 + (-3.5 + 0.866i)T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6 - 3.46i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.66iT - 47T^{2} \) |
| 53 | \( 1 - 8.66iT - 53T^{2} \) |
| 59 | \( 1 + (-9 + 5.19i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 + (7.5 + 4.33i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.82318833695876405541139087297, −9.507122661028961176946428581338, −8.954642786528935692383632500401, −8.399518132699041492276816647180, −7.29334407394395084745429439575, −5.35723241514589119346424560334, −4.85466274263137514455950931304, −4.07531622514595584239566222180, −2.55794422222255809473022854115, −1.40282037575030791985896656203,
1.94506129643303569782553429207, 3.28623842475651424666593322353, 3.98626857949496222011960525997, 5.67233096032637178991782036792, 6.66808747737944856848212619749, 7.20487540778183537262662955882, 8.358243979180356856401867568068, 8.583335330917070372761100024260, 10.30765734111193944929865928702, 10.84177431309224925357041016537