L(s) = 1 | + (−1.19 + 0.769i)2-s + (0.989 + 0.142i)3-s + (0.0103 − 0.0227i)4-s + (−1.85 + 0.543i)5-s + (−1.29 + 0.591i)6-s + (2.00 − 1.72i)7-s + (−0.399 − 2.78i)8-s + (0.959 + 0.281i)9-s + (1.79 − 2.07i)10-s + (−1.95 + 3.04i)11-s + (0.0135 − 0.0210i)12-s + (2.98 + 2.58i)13-s + (−1.07 + 3.60i)14-s + (−1.90 + 0.274i)15-s + (2.65 + 3.06i)16-s + (2.49 + 5.45i)17-s + ⋯ |
L(s) = 1 | + (−0.846 + 0.544i)2-s + (0.571 + 0.0821i)3-s + (0.00519 − 0.0113i)4-s + (−0.827 + 0.242i)5-s + (−0.528 + 0.241i)6-s + (0.758 − 0.651i)7-s + (−0.141 − 0.983i)8-s + (0.319 + 0.0939i)9-s + (0.568 − 0.655i)10-s + (−0.590 + 0.918i)11-s + (0.00390 − 0.00607i)12-s + (0.827 + 0.717i)13-s + (−0.287 + 0.964i)14-s + (−0.492 + 0.0708i)15-s + (0.662 + 0.765i)16-s + (0.604 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.315709 + 0.729959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.315709 + 0.729959i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.989 - 0.142i)T \) |
| 7 | \( 1 + (-2.00 + 1.72i)T \) |
| 23 | \( 1 + (4.57 + 1.44i)T \) |
good | 2 | \( 1 + (1.19 - 0.769i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (1.85 - 0.543i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (1.95 - 3.04i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.98 - 2.58i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.49 - 5.45i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.11 - 4.62i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.273 + 0.599i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (9.09 - 1.30i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.49 + 5.10i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.999 + 3.40i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-11.5 - 1.65i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 8.64iT - 47T^{2} \) |
| 53 | \( 1 + (4.59 - 3.98i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.75 - 2.39i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 8.21i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (2.51 + 3.92i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.52 + 1.62i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.80 - 0.824i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.73 - 7.56i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.39 - 0.410i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.896 - 6.23i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-9.51 + 2.79i)T + (81.6 - 52.4i)T^{2} \) |
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
−10.94727931631280125726699508600, −10.35389357494811315658867454084, −9.323904596694091729801111732581, −8.350304287001042325575677794495, −7.75780519571154132441606601306, −7.32413219797697031346052437946, −6.00868176128450113674282452074, −4.08846077531618791838968700486, −3.86769500990330959345066902038, −1.72191306532352271631635644543,
0.61953603164120456026797026710, 2.23847415182015011807538487328, 3.39776929641527341811805856068, 4.92634341730601749158194309503, 5.80821757495002240533354415634, 7.60603299337833489289886122200, 8.185662403277643159483227789876, 8.785740109099573799466845601456, 9.603185887310901434027930952223, 10.77689412930717880758377881320