L(s) = 1 | − 1.73i·3-s + (−1.5 + 0.866i)5-s + (2 − 1.73i)7-s − 2.99·9-s + (4.5 + 2.59i)11-s + (4.5 + 2.59i)13-s + (1.49 + 2.59i)15-s + (1.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−2.99 − 3.46i)21-s + (−1.5 + 0.866i)23-s + (−1 + 1.73i)25-s + 5.19i·27-s + (−4.5 − 7.79i)29-s + 8·31-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + (−0.670 + 0.387i)5-s + (0.755 − 0.654i)7-s − 0.999·9-s + (1.35 + 0.783i)11-s + (1.24 + 0.720i)13-s + (0.387 + 0.670i)15-s + (0.363 − 0.210i)17-s + (−0.114 + 0.198i)19-s + (−0.654 − 0.755i)21-s + (−0.312 + 0.180i)23-s + (−0.200 + 0.346i)25-s + 0.999i·27-s + (−0.835 − 1.44i)29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.685443052\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.685443052\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 0.866i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 0.866i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.5 + 4.33i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.5 + 2.59i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.793729036322999241866783382965, −8.864336124553825201122839990937, −7.956148595561574274328642057049, −7.39489535843039461628565915793, −6.62701813282801079196907969127, −5.83426546936073201793595869096, −4.27560390998925772040728135241, −3.73489176492858467142154619881, −2.07757653680956869719052711680, −1.05807239814880289810880782296,
1.15152456930954724275369187688, 3.03962269942125000751799917312, 3.89517010216574636004359925541, 4.68962662849578291443718416228, 5.71967155882627164162479305166, 6.39455468734440436651062536442, 8.098498096673459878601758301295, 8.413204026205031106887681993222, 9.079501695371213787237513903654, 10.05814689180805677664821983439