L(s) = 1 | − i·2-s + (−0.765 + 1.55i)3-s − 4-s + (−0.474 − 0.821i)5-s + (1.55 + 0.765i)6-s + i·8-s + (−1.82 − 2.37i)9-s + (−0.821 + 0.474i)10-s + (−1.51 − 0.876i)11-s + (0.765 − 1.55i)12-s + (0.720 + 0.416i)13-s + (1.64 − 0.108i)15-s + 16-s + (2.73 + 4.73i)17-s + (−2.37 + 1.82i)18-s + (3.21 + 1.85i)19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.441 + 0.897i)3-s − 0.5·4-s + (−0.212 − 0.367i)5-s + (0.634 + 0.312i)6-s + 0.353i·8-s + (−0.609 − 0.792i)9-s + (−0.259 + 0.150i)10-s + (−0.457 − 0.264i)11-s + (0.220 − 0.448i)12-s + (0.199 + 0.115i)13-s + (0.423 − 0.0280i)15-s + 0.250·16-s + (0.663 + 1.14i)17-s + (−0.560 + 0.431i)18-s + (0.736 + 0.425i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.827043 + 0.433135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.827043 + 0.433135i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.765 - 1.55i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.474 + 0.821i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.51 + 0.876i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.720 - 0.416i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.73 - 4.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.21 - 1.85i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.888 + 0.513i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.58 - 3.80i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 11.0iT - 31T^{2} \) |
| 37 | \( 1 + (2.19 - 3.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.05 - 7.02i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.08 - 5.34i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 + (-1.88 + 1.08i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 1.09iT - 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 11.0iT - 71T^{2} \) |
| 73 | \( 1 + (3.16 - 1.82i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 9.84T + 79T^{2} \) |
| 83 | \( 1 + (2.41 + 4.17i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.15 + 3.74i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.90 - 2.25i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.30681531187440103565437380417, −9.706915223742583873610089516561, −8.670692591390358543946982239366, −8.153994356233221753307206330535, −6.67470293551033336995210566703, −5.53392009563744699873022669162, −4.92474606633627251037988661848, −3.81244570375924873439441435428, −3.10080270282477412305021659537, −1.27312316745455171088028160419,
0.53633316413634416935545101922, 2.31007304217632136780653232126, 3.62660875857034468366093385069, 5.15177978616590920221442358315, 5.61476433391505415882671997289, 6.76516511665313501826445943812, 7.44340288081499179658067718134, 7.86085061420820530511739274168, 9.068939710794885484597314749197, 9.886849443020991188398441395955