L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−1.87 − 1.08i)5-s − 0.999i·6-s + (2.59 + 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.08 − 1.87i)10-s + (3.60 − 2.08i)11-s + (0.499 − 0.866i)12-s + (−0.418 − 3.58i)13-s + (2 + 1.73i)14-s + 2.16i·15-s + (−0.5 + 0.866i)16-s + (−2.58 − 4.47i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.837 − 0.483i)5-s − 0.408i·6-s + (0.981 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.341 − 0.592i)10-s + (1.08 − 0.627i)11-s + (0.144 − 0.249i)12-s + (−0.116 − 0.993i)13-s + (0.534 + 0.462i)14-s + 0.558i·15-s + (−0.125 + 0.216i)16-s + (−0.626 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77040 - 0.484180i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77040 - 0.484180i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.59 - 0.5i)T \) |
| 13 | \( 1 + (0.418 + 3.58i)T \) |
good | 5 | \( 1 + (1.87 + 1.08i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.60 + 2.08i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.58 + 4.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.47 - 3.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.581 - 1.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 + (-7.79 + 4.5i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.47 - 3.16i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 + (11.6 + 6.74i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.08 - 5.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.59 + 0.918i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.74 - 13.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.45 - 1.41i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.67iT - 71T^{2} \) |
| 73 | \( 1 + (5.47 - 3.16i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.8iT - 83T^{2} \) |
| 89 | \( 1 + (2.73 + 1.58i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11iT - 97T^{2} \) |
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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
−11.25995524568514572132948117499, −9.847844752814316370028776873394, −8.456342085186199950945989001287, −8.060836690040475970889444463924, −7.13382367150558017519945572055, −6.04043927382150195265575607498, −5.09783353197850137469182577216, −4.25246137469006140162273348245, −2.93864133378320066658680657971, −1.06926384415133234537237377327,
1.61450229313677943899627866090, 3.29529374213329812623935324795, 4.37889410177478570951439802166, 4.76070527888434855835904205546, 6.34567421083898737907045923652, 7.05170495825813262513171575650, 8.197358011826424234882027350757, 9.283914888928450409299044448007, 10.22350814975159682234288660492, 11.22382896918745173262381290678