L(s) = 1 | + (−0.966 − 0.257i)2-s + (−0.166 − 0.985i)3-s + (0.867 + 0.498i)4-s + (0.626 − 0.779i)5-s + (−0.0929 + 0.995i)6-s + (−0.616 − 0.787i)7-s + (−0.709 − 0.704i)8-s + (−0.944 + 0.329i)9-s + (−0.806 + 0.591i)10-s + (0.767 + 0.640i)11-s + (0.346 − 0.938i)12-s + (0.975 − 0.221i)13-s + (0.392 + 0.919i)14-s + (−0.873 − 0.487i)15-s + (0.503 + 0.863i)16-s + (0.700 − 0.713i)17-s + ⋯ |
L(s) = 1 | + (−0.966 − 0.257i)2-s + (−0.166 − 0.985i)3-s + (0.867 + 0.498i)4-s + (0.626 − 0.779i)5-s + (−0.0929 + 0.995i)6-s + (−0.616 − 0.787i)7-s + (−0.709 − 0.704i)8-s + (−0.944 + 0.329i)9-s + (−0.806 + 0.591i)10-s + (0.767 + 0.640i)11-s + (0.346 − 0.938i)12-s + (0.975 − 0.221i)13-s + (0.392 + 0.919i)14-s + (−0.873 − 0.487i)15-s + (0.503 + 0.863i)16-s + (0.700 − 0.713i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3116667808 - 0.8709808090i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3116667808 - 0.8709808090i\) |
\(L(1)\) |
\(\approx\) |
\(0.6003113936 - 0.4906294694i\) |
\(L(1)\) |
\(\approx\) |
\(0.6003113936 - 0.4906294694i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.966 - 0.257i)T \) |
| 3 | \( 1 + (-0.166 - 0.985i)T \) |
| 5 | \( 1 + (0.626 - 0.779i)T \) |
| 7 | \( 1 + (-0.616 - 0.787i)T \) |
| 11 | \( 1 + (0.767 + 0.640i)T \) |
| 13 | \( 1 + (0.975 - 0.221i)T \) |
| 17 | \( 1 + (0.700 - 0.713i)T \) |
| 19 | \( 1 + (0.999 - 0.0248i)T \) |
| 29 | \( 1 + (-0.556 - 0.831i)T \) |
| 31 | \( 1 + (-0.492 - 0.870i)T \) |
| 37 | \( 1 + (0.890 - 0.454i)T \) |
| 41 | \( 1 + (0.735 + 0.678i)T \) |
| 43 | \( 1 + (0.545 + 0.837i)T \) |
| 47 | \( 1 + (-0.334 - 0.942i)T \) |
| 53 | \( 1 + (-0.806 - 0.591i)T \) |
| 59 | \( 1 + (0.827 + 0.561i)T \) |
| 61 | \( 1 + (-0.999 + 0.0372i)T \) |
| 67 | \( 1 + (0.992 + 0.123i)T \) |
| 71 | \( 1 + (0.922 + 0.386i)T \) |
| 73 | \( 1 + (-0.556 + 0.831i)T \) |
| 79 | \( 1 + (-0.287 - 0.957i)T \) |
| 83 | \( 1 + (-0.820 + 0.571i)T \) |
| 89 | \( 1 + (-0.834 - 0.551i)T \) |
| 97 | \( 1 + (-0.514 + 0.857i)T \) |
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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
−23.89218926027467882298044755276, −22.79690467788686623189487919914, −21.98314619431703128452877621281, −21.37984998665807664547204176578, −20.45456138387383925881840874189, −19.40654392459539786408764536608, −18.66469652548883929802550974652, −17.9292515570021605125596965495, −16.94696772696919318842609150705, −16.233736007717496206362704240515, −15.561162289556503740791958330197, −14.610529302046877167404494767680, −13.998512725169578669112953893111, −12.32759698504054596405678896058, −11.2223331477931058010372279489, −10.753417108559515488073231912520, −9.63382250426948304321644993847, −9.240028766217125066582400556, −8.32038353380365376014907579327, −6.86082827737797673689189691368, −5.9517256583686712636311419029, −5.57886506980271570104833228889, −3.57449007710918768124653153714, −2.91267896405936096485651278772, −1.43587045504021682548734087645,
0.79649197692600037310814686155, 1.42353370546134903628549901534, 2.66329901380013427552708295416, 3.93197510061940484301374636312, 5.64963661042604305644980951087, 6.451496971114758648747914199067, 7.38692579017737034076741275109, 8.11717257416472955875508866869, 9.37670289974744045010164557441, 9.736884700444320743827687165951, 11.09904991749594488742797239667, 11.867377684803487445013401703579, 12.836992578334293014864795693618, 13.37435654386604545852381434790, 14.44066922917256811238000668588, 16.05522682862756793647159927187, 16.62317658601088788592765332475, 17.36098734510732613354626257864, 18.06602348220456250203712940599, 18.82150419982700530639718691493, 19.967398166143403176877421790844, 20.201469635337936222201917746927, 21.11394483925069137418596641156, 22.471315424500874438526935800291, 23.17811426085736208510016298301