L(s) = 1 | + (−1.5 − 0.866i)3-s + (1 + 1.73i)5-s + (1.5 + 2.59i)9-s + (−2 + 3.46i)11-s + (1.5 + 2.59i)13-s − 3.46i·15-s − 7·17-s − 5·19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s − 5.19i·27-s + (0.5 − 0.866i)29-s + (−1.5 − 2.59i)31-s + (6 − 3.46i)33-s + 11·37-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (0.447 + 0.774i)5-s + (0.5 + 0.866i)9-s + (−0.603 + 1.04i)11-s + (0.416 + 0.720i)13-s − 0.894i·15-s − 1.69·17-s − 1.14·19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s − 0.999i·27-s + (0.0928 − 0.160i)29-s + (−0.269 − 0.466i)31-s + (1.04 − 0.603i)33-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.964419060660366002131234864032, −8.016936254678466207074755604218, −7.08375758753444117725567138885, −6.50298383934735771135948085540, −6.04788214020341261428847963889, −4.74107154704612614656140232953, −4.24015753110969675349307242265, −2.41390891911840451304352225335, −1.97221232381004059382185293125, 0,
1.31596483718426853818580891771, 2.82680109581761508252724427473, 4.06731067756009020546767167620, 4.78864368759123323322049562980, 5.71560435912110414299267860097, 6.09425771595818714518119591384, 7.12685167474795544954305853700, 8.401828442189882577144039273305, 8.789897191943840195728656688343