L(s) = 1 | + (−0.5 + 0.866i)5-s + (−2 − 1.73i)7-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 − 0.866i)35-s + (2.5 − 4.33i)37-s + 10·41-s + 4·43-s + (−0.5 + 0.866i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (−0.755 − 0.654i)7-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 + (0.422 − 0.146i)35-s + (0.410 − 0.711i)37-s + 1.56·41-s + 0.609·43-s + (−0.0729 + 0.126i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.469402574\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469402574\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + 1.73i)T \) |
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
−9.916513076113166673311189167055, −9.248031571722641190187161665819, −8.255758357352845241190116196515, −7.27329794052968321813846905390, −6.70305336083795668527057368425, −5.77885638114771607223014297443, −4.53257981401221815748613933892, −3.62435477072564995386089615425, −2.69567004105066359879896635740, −0.929291784208924058092353344144,
1.02886509453587497814523678664, 2.70611673683731409588438346324, 3.59938462516958460735302982934, 4.76554360252215029645398792202, 5.78584299680618383915517211986, 6.43074901407398062327640143470, 7.57333318884158429398472728881, 8.384863079289364299930103497260, 9.190973368019776803627160521415, 9.822619217327604104467481483554