L(s) = 1 | + (1.20 − 2.09i)2-s + (−1.91 − 3.31i)4-s + (−1.41 − 2.44i)5-s + (1.41 − 2.44i)7-s − 4.41·8-s − 6.82·10-s + (1 − 1.73i)11-s + (0.5 + 0.866i)13-s + (−3.41 − 5.91i)14-s + (−1.49 + 2.59i)16-s − 3.65·17-s + 2.82·19-s + (−5.41 + 9.37i)20-s + (−2.41 − 4.18i)22-s + (2 + 3.46i)23-s + ⋯ |
L(s) = 1 | + (0.853 − 1.47i)2-s + (−0.957 − 1.65i)4-s + (−0.632 − 1.09i)5-s + (0.534 − 0.925i)7-s − 1.56·8-s − 2.15·10-s + (0.301 − 0.522i)11-s + (0.138 + 0.240i)13-s + (−0.912 − 1.58i)14-s + (−0.374 + 0.649i)16-s − 0.886·17-s + 0.648·19-s + (−1.21 + 2.09i)20-s + (−0.514 − 0.891i)22-s + (0.417 + 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.120585575\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120585575\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.41 + 2.44i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.41 + 2.44i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.41 - 5.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + (5.41 + 9.37i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.82 - 8.36i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.171 + 0.297i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (-1.82 - 3.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.65 + 8.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.585 + 1.01i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + (5.65 - 9.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.82 + 6.63i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 + (-3.82 + 6.63i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.584952066479522763368681936085, −8.800678384327272359883352881690, −7.936401325871520443129849173635, −6.80348239683497819347426909173, −5.36948367860488835685257758147, −4.69974980131245745962409037361, −4.03277231630527918776244654718, −3.20912441611431608167577365191, −1.63889493769614887654267736238, −0.77244955387151828656093101702,
2.44534883060535764075915599123, 3.58078653369610791702588306013, 4.53728428351395401629707022482, 5.33834253117311400815916410242, 6.37062448759988580936562266392, 6.85062812474068281997134737709, 7.74984562626546365706673013643, 8.322821904495593475090486032213, 9.268621873159362624884828466082, 10.46835972961785291747595187687