L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 0.999·6-s + (−2.59 − 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.5 − 4.33i)11-s + (−0.866 − 0.499i)12-s + (−2 − 1.73i)14-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s + (0.866 − 0.499i)18-s + (4 − 6.92i)19-s + (2.5 − 0.866i)21-s − 5i·22-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s − 0.408·6-s + (−0.981 − 0.188i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.753 − 1.30i)11-s + (−0.249 − 0.144i)12-s + (−0.534 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s + (0.204 − 0.117i)18-s + (0.917 − 1.58i)19-s + (0.545 − 0.188i)21-s − 1.06i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.399032680\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.399032680\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.59 + 0.5i)T \) |
good | 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.46 + 2i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (-8.66 + 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7iT - 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−9.787686932739782184045027062756, −9.106938615991996137213939911630, −7.988824873398925967319477934472, −7.09203633572202663977570051210, −6.33117746254756987682754920623, −5.45291232493097690878108713892, −4.83927145360409769722314477828, −3.39961154514182652700317008574, −2.95418766513610313660702410090, −0.57234456117050874678171243445,
1.42711203661123366459042099296, 2.73979647377619173140643440214, 3.71571472285261648147546175966, 4.92436881377592436431929597252, 5.62078783408022173939881494122, 6.52152700919763019293472512045, 7.29317497113806756899872891544, 8.205218598157456482141996757218, 9.597991056064443436478238890433, 10.10269567857564702414554259797