L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s − 2·13-s + (−1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (−1.5 − 2.59i)23-s + (2 − 3.46i)25-s + 6·29-s + (−0.5 + 0.866i)31-s + (2.5 + 4.33i)37-s − 10·41-s − 4·43-s + (−0.5 − 0.866i)47-s + (−4.5 + 7.79i)53-s − 0.999·55-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (−0.150 + 0.261i)11-s − 0.554·13-s + (−0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (−0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s + 1.11·29-s + (−0.0898 + 0.155i)31-s + (0.410 + 0.711i)37-s − 1.56·41-s − 0.609·43-s + (−0.0729 − 0.126i)47-s + (−0.618 + 1.07i)53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.110763939\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110763939\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.677264777718067008888600378818, −8.166423310383080954810556587018, −7.31726766036183524478256079266, −6.56202825699481683822786721348, −5.97523649418232971322541172816, −4.97960367384078813936593535257, −4.29600244192821766558779229927, −3.23116615740893548146609606076, −2.43213559524387488486262845474, −1.36663520754116046331529596047,
0.32665185366113103000616308488, 1.60678278311467160890563830409, 2.71244141787279812490716823366, 3.48890427283877296543872776650, 4.79613761986216462137133710896, 5.03798883183762878239337995928, 6.05568213534880117769942171714, 6.90607958541045003457299432342, 7.50869874515317231007050691188, 8.384801328355182377862302508174