L(s) = 1 | + (0.5 + 0.866i)2-s + (1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−2.63 + 1.52i)5-s + 1.73i·6-s + (2.63 + 0.209i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−2.63 − 1.52i)10-s + (−2.5 − 2.17i)11-s + (−1.49 + 0.866i)12-s + 6.09i·13-s + (1.13 + 2.38i)14-s − 5.27·15-s + (−0.5 − 0.866i)16-s + (2.13 − 3.70i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−1.17 + 0.680i)5-s + 0.707i·6-s + (0.996 + 0.0791i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.834 − 0.481i)10-s + (−0.753 − 0.657i)11-s + (−0.433 + 0.250i)12-s + 1.68i·13-s + (0.303 + 0.638i)14-s − 1.36·15-s + (−0.125 − 0.216i)16-s + (0.518 − 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.763479 + 1.66566i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.763479 + 1.66566i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.63 - 0.209i)T \) |
| 11 | \( 1 + (2.5 + 2.17i)T \) |
good | 5 | \( 1 + (2.63 - 1.52i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 6.09iT - 13T^{2} \) |
| 17 | \( 1 + (-2.13 + 3.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.86 + 1.07i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.41 - 3.70i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (-1.36 + 2.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.137 + 0.238i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.54T + 41T^{2} \) |
| 43 | \( 1 + 5.61iT - 43T^{2} \) |
| 47 | \( 1 + (-9.41 + 5.43i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.18 - 4.14i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 0.866i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.27 - 5.67i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.61iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 + 6.09i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.63 + 3.25i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 + (-5.27 + 3.04i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.54T + 97T^{2} \) |
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
−11.53247770494693660082073505585, −10.55433473761013623881278162099, −9.329900777230092772898113814494, −8.450965159251886546499944423362, −7.62238238428164701709358103708, −7.21914506834317024620031830564, −5.56511363523310382330133237298, −4.41799023783849148920457403516, −3.73658488115899998778219377053, −2.47021354102504723268699959946,
0.998232693544978054826705476656, 2.50248424806857233177714434695, 3.76848694640105692047849025268, 4.60663407343072318202210633098, 5.77180353286523159328634888180, 7.63027171604852118117535807468, 7.942126861515934807031836336918, 8.626487480845458607156033180008, 10.02261744788417784326807967152, 10.71502874946255934764456885269