L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.286 + 0.957i)3-s + (−0.939 + 0.342i)4-s + (0.0581 + 0.998i)5-s + (0.993 + 0.116i)6-s + (0.893 + 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.0581 − 0.998i)12-s + (−0.893 − 0.448i)13-s + (0.286 − 0.957i)14-s + (−0.973 − 0.230i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.286 + 0.957i)3-s + (−0.939 + 0.342i)4-s + (0.0581 + 0.998i)5-s + (0.993 + 0.116i)6-s + (0.893 + 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.0581 − 0.998i)12-s + (−0.893 − 0.448i)13-s + (0.286 − 0.957i)14-s + (−0.973 − 0.230i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.113079337 - 0.2631741855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.113079337 - 0.2631741855i\) |
\(L(1)\) |
\(\approx\) |
\(0.8513756112 + 0.01456374892i\) |
\(L(1)\) |
\(\approx\) |
\(0.8513756112 + 0.01456374892i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.286 + 0.957i)T \) |
| 5 | \( 1 + (0.0581 + 0.998i)T \) |
| 7 | \( 1 + (0.893 + 0.448i)T \) |
| 11 | \( 1 + (0.973 + 0.230i)T \) |
| 13 | \( 1 + (-0.893 - 0.448i)T \) |
| 17 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.686 + 0.727i)T \) |
| 31 | \( 1 + (0.0581 - 0.998i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.396 - 0.918i)T \) |
| 53 | \( 1 + (0.893 + 0.448i)T \) |
| 59 | \( 1 + (-0.973 - 0.230i)T \) |
| 61 | \( 1 + (0.993 + 0.116i)T \) |
| 67 | \( 1 + (-0.835 - 0.549i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.973 + 0.230i)T \) |
| 79 | \( 1 + (0.0581 - 0.998i)T \) |
| 83 | \( 1 + (0.973 - 0.230i)T \) |
| 89 | \( 1 + (-0.396 - 0.918i)T \) |
| 97 | \( 1 + (-0.893 - 0.448i)T \) |
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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
−18.21628712278468454619609585941, −17.55605522371812658691060956830, −17.29774990925724023247354754206, −16.59340005762676473592445755290, −16.11411520708135985650960342405, −15.07198403721117850749280628556, −14.10071260293858927115164049355, −14.02861143909153843025932477591, −13.27079989352112302097439080936, −12.264467278655922354375522897105, −11.963785626867290638132018624710, −11.07840028544296981062115181981, −9.99733220191646695057593991701, −9.26852643453299892458159170489, −8.50243653323771173391571078815, −8.03256529569292759140521974328, −7.32454576908248822234248995573, −6.66732730862824539517613446717, −5.907194991982682837350304622568, −5.180561935890786663730119384691, −4.53049559648491865502741701467, −3.88147828173657193586594934688, −2.23395488914089932138998029143, −1.40377634936888580900072948515, −0.82446559205888330450217623155,
0.473359042273017754529531940308, 1.94977336716252886493122594483, 2.44644130270770285138699494058, 3.28933202448812432495215792, 4.201348590853535689642534351727, 4.58640860421964941977000743003, 5.49758516587325559564898661808, 6.32197597799811752459628532445, 7.294107600975248890705252552460, 8.260429019752801522438749209393, 8.95372706977899740470692854992, 9.518285196982822254533696521621, 10.48817386678237665936114442291, 10.63266336695034168563537223838, 11.52711933177957284467252492962, 11.897643916412406163977416827822, 12.652685529142917008977806711752, 13.8835727957057803242707958588, 14.29363846524723178246962286235, 15.14147554574137674189093410174, 15.27289597736013839417066582618, 16.658101452517180745749924123760, 17.345731484086788481804673799502, 17.73499822500573940340676220860, 18.274730125576940961915987427773