L(s) = 1 | + 1.73i·3-s + (−0.5 + 0.866i)5-s + (−1.5 − 2.59i)7-s − 2.99·9-s + (−2.5 − 4.33i)11-s + (−2.5 + 4.33i)13-s + (−1.49 − 0.866i)15-s − 2·17-s − 4·19-s + (4.5 − 2.59i)21-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s − 5.19i·27-s + (−4.5 − 7.79i)29-s + (−0.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (−0.223 + 0.387i)5-s + (−0.566 − 0.981i)7-s − 0.999·9-s + (−0.753 − 1.30i)11-s + (−0.693 + 1.20i)13-s + (−0.387 − 0.223i)15-s − 0.485·17-s − 0.917·19-s + (0.981 − 0.566i)21-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s − 0.999i·27-s + (−0.835 − 1.44i)29-s + (−0.0898 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.43492694304869057440819661767, −9.625237058287413707032893783758, −8.785961773662585763219100354660, −7.75047630207729020276856840071, −6.71086566929469028540821028833, −5.76532420064636348290992978902, −4.48240781200315362118044152200, −3.74771966390283986014329240191, −2.64620976574182958188337782782, 0,
2.07782862536966958638799869133, 2.93162622077891326075837830644, 4.73935065587436680190402379325, 5.61528410610409978086240987530, 6.62603027517487974904034472353, 7.54395960506447038417362647672, 8.310401263809879282965846006096, 9.170502737496770101100013179605, 10.16818066743046897339602178808