L(s) = 1 | + (−0.826 + 0.563i)3-s + (0.900 + 0.433i)5-s + (0.365 − 0.930i)9-s + (0.988 + 0.149i)11-s + (0.365 + 0.930i)13-s + (−0.988 + 0.149i)15-s + (0.733 + 0.680i)17-s + (0.733 − 0.680i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)27-s + (0.733 + 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.900 + 0.433i)33-s + (0.733 + 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.563i)3-s + (0.900 + 0.433i)5-s + (0.365 − 0.930i)9-s + (0.988 + 0.149i)11-s + (0.365 + 0.930i)13-s + (−0.988 + 0.149i)15-s + (0.733 + 0.680i)17-s + (0.733 − 0.680i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)27-s + (0.733 + 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.900 + 0.433i)33-s + (0.733 + 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.127022593 + 2.445081225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.127022593 + 2.445081225i\) |
\(L(1)\) |
\(\approx\) |
\(1.065943431 + 0.5069646529i\) |
\(L(1)\) |
\(\approx\) |
\(1.065943431 + 0.5069646529i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.826 + 0.563i)T \) |
| 5 | \( 1 + (0.900 + 0.433i)T \) |
| 11 | \( 1 + (0.988 + 0.149i)T \) |
| 13 | \( 1 + (0.365 + 0.930i)T \) |
| 17 | \( 1 + (0.733 + 0.680i)T \) |
| 23 | \( 1 + (0.733 - 0.680i)T \) |
| 29 | \( 1 + (0.733 + 0.680i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.733 + 0.680i)T \) |
| 41 | \( 1 + (0.0747 + 0.997i)T \) |
| 43 | \( 1 + (-0.826 - 0.563i)T \) |
| 47 | \( 1 + (-0.988 - 0.149i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 + (-0.0747 + 0.997i)T \) |
| 61 | \( 1 + (-0.955 + 0.294i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.733 + 0.680i)T \) |
| 73 | \( 1 + (0.988 - 0.149i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (0.365 - 0.930i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.95252817318273813582672852948, −17.59798821694882944234232303021, −16.91320073843135993310727708708, −16.42873390522102143025243798854, −15.627151336922219859296791491662, −14.670596329811669799515348525600, −13.85301790833040045967054236811, −13.350040146487878994197756620431, −12.71322598101286464344082959079, −11.99093846815209992288415669315, −11.38339986688111012189875355133, −10.611176502025346025736716675742, −9.78944211226354966093197955350, −9.298205654183701671346111737013, −8.221650734111645124355595741606, −7.64897898916835444346820296647, −6.556093894491911344305895977122, −6.22365108187526154993505071314, −5.338314196881242742343088979110, −4.933907426266720292731105947598, −3.79695053684220760886632812778, −2.77323511911442314020131840032, −1.83861331564186634045298819260, −1.01003168159270225647701039633, −0.5305560543795357072335565790,
1.12446697909096164694105599703, 1.48656889174035892287342400918, 2.80814003575284219059076263324, 3.59401390692849079880560291923, 4.47448291869496146786208599346, 5.102193180941456134728612897687, 6.05641823030366643349017317620, 6.544116424822816914657009895856, 6.977340770863643734606006755872, 8.37166346360736831406347900177, 9.13182360749966456720461328683, 9.70907242680076012875625969582, 10.41099805349838254782023602485, 10.94467091243972872857333926896, 11.78422349464956814161592692751, 12.30337048971785910004084500744, 13.19508267578837449946641004263, 14.03090967204969548835503809568, 14.66611183540464481309992315682, 15.13437111714497731274522629292, 16.23351067003156894982172432253, 16.80095148677385610207637778668, 17.115286076670796836268673626797, 18.02565258361448569856004276974, 18.46685685262840038657383221412