L(s) = 1 | + (1 + 1.73i)4-s + (−2 + 1.73i)7-s + (−1 + 1.73i)13-s + (−1.99 + 3.46i)16-s + (3.5 + 6.06i)19-s − 5·25-s + (−5 − 1.73i)28-s + (−5.5 − 9.52i)31-s + (5 + 8.66i)37-s + (6.5 + 11.2i)43-s + (1.00 − 6.92i)49-s − 3.99·52-s + (6.5 − 11.2i)61-s − 7.99·64-s + (8 + 13.8i)67-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)4-s + (−0.755 + 0.654i)7-s + (−0.277 + 0.480i)13-s + (−0.499 + 0.866i)16-s + (0.802 + 1.39i)19-s − 25-s + (−0.944 − 0.327i)28-s + (−0.987 − 1.71i)31-s + (0.821 + 1.42i)37-s + (0.991 + 1.71i)43-s + (0.142 − 0.989i)49-s − 0.554·52-s + (0.832 − 1.44i)61-s − 0.999·64-s + (0.977 + 1.69i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.757424 + 1.02851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.757424 + 1.02851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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
−11.33754783024801363119247640029, −9.911788745813873248605678885613, −9.385546361136600896424394363798, −8.179289221383141528166678456506, −7.56466893603838790489181002051, −6.44893484596486964282560594211, −5.71772194824865225680159307629, −4.15697473406131112685907163469, −3.20271871530117632268876883518, −2.08312097283496730646427202482,
0.70416760303996808035751456476, 2.39363743426785320378985176239, 3.64447261449734748074486177092, 5.04396931776772678376488569330, 5.88012408238237531953535425725, 6.98324181287059409544472946414, 7.44097026221446841493736443232, 9.007718827836587567958558466615, 9.687850839425261836008690189877, 10.53384890986988536731388267521