L(s) = 1 | + (0.5 + 0.866i)2-s + (0.619 − 1.61i)3-s + (−0.499 + 0.866i)4-s + (−1.59 + 2.75i)5-s + (1.71 − 0.272i)6-s − 0.999·8-s + (−2.23 − 2.00i)9-s − 3.18·10-s + (−1.59 − 2.75i)11-s + (1.09 + 1.34i)12-s + (−2.85 + 4.93i)13-s + (3.47 + 4.28i)15-s + (−0.5 − 0.866i)16-s − 1.52·17-s + (0.619 − 2.93i)18-s − 1.28·19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.357 − 0.933i)3-s + (−0.249 + 0.433i)4-s + (−0.711 + 1.23i)5-s + (0.698 − 0.111i)6-s − 0.353·8-s + (−0.744 − 0.668i)9-s − 1.00·10-s + (−0.479 − 0.830i)11-s + (0.314 + 0.388i)12-s + (−0.790 + 1.36i)13-s + (0.896 + 1.10i)15-s + (−0.125 − 0.216i)16-s − 0.369·17-s + (0.146 − 0.691i)18-s − 0.294·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00882256 + 0.526913i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00882256 + 0.526913i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.619 + 1.61i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.59 - 2.75i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.85 - 4.93i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + (1.11 - 1.93i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.71 - 8.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 + (0.562 - 0.974i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.56 - 2.70i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.69T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 - 6.98i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 0.225T + 89T^{2} \) |
| 97 | \( 1 + (7.42 + 12.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.80303558538234928949019964571, −9.474691883173347633635106713020, −8.582347397632892340857827451008, −7.72892013012262049871044440148, −7.09760411221788208643661980369, −6.56185567469204515968154738936, −5.58130357992968394428316705897, −4.13041015119845681670262599970, −3.20909594041700943722037861058, −2.21361231639694213251103570002,
0.19878543287695356164102709028, 2.18468639422949175003931924685, 3.37024481652465288313903189158, 4.40000979252967198994723719982, 4.91478578041361297712965995499, 5.70380712307994137359587174269, 7.52918457935349678126897742996, 8.166564844498692937028640666992, 9.093887721988965283285654096177, 9.720676249980402266599786939067