L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 2·11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + 0.999·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + 0.408·6-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.603·11-s + (−0.144 + 0.249i)12-s + (0.138 + 0.240i)13-s + 0.267·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5651267782\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5651267782\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (3.76 + 4.78i)T \) |
good | 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.76 + 4.78i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.52 + 7.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.260 + 0.451i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.239 - 0.415i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + (1.26 + 2.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.23 + 3.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.52 + 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.52T + 97T^{2} \) |
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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
−9.592196004880319674960385645199, −8.634273178213061539872369856757, −7.87758550025610728433955749827, −6.89475143073441270285859757783, −6.59416362851541573609678281181, −5.46293356998470165968778978431, −4.67571292449856941092535622557, −3.20808306918801663078071290235, −1.92728931955661281266567042587, −0.29108407088561789956321628625,
1.48497929366756751508745332580, 2.81499334434102648868542469132, 3.82729047466218231111988019509, 4.86674544262026924553633095874, 5.69927593445464258495301215574, 6.63471480118783950062696227810, 8.013236192130564381053978538424, 8.498753944060611820911770812890, 9.417265629494266173976187489947, 10.27504376082615526100892068597