L(s) = 1 | + (0.923 + 0.382i)3-s + (−0.831 − 0.555i)5-s + (0.831 − 0.555i)7-s + (0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.555 − 0.831i)13-s + (−0.555 − 0.831i)15-s + (0.555 − 0.831i)17-s + (−0.831 − 0.555i)19-s + (0.980 − 0.195i)21-s + (0.195 + 0.980i)23-s + (0.382 + 0.923i)25-s + (0.382 + 0.923i)27-s + (−0.195 − 0.980i)29-s + (0.382 − 0.923i)31-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)3-s + (−0.831 − 0.555i)5-s + (0.831 − 0.555i)7-s + (0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.555 − 0.831i)13-s + (−0.555 − 0.831i)15-s + (0.555 − 0.831i)17-s + (−0.831 − 0.555i)19-s + (0.980 − 0.195i)21-s + (0.195 + 0.980i)23-s + (0.382 + 0.923i)25-s + (0.382 + 0.923i)27-s + (−0.195 − 0.980i)29-s + (0.382 − 0.923i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526378682 - 0.5704107455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526378682 - 0.5704107455i\) |
\(L(1)\) |
\(\approx\) |
\(1.315562695 - 0.1825464940i\) |
\(L(1)\) |
\(\approx\) |
\(1.315562695 - 0.1825464940i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 97 | \( 1 \) |
good | 3 | \( 1 + (0.923 + 0.382i)T \) |
| 5 | \( 1 + (-0.831 - 0.555i)T \) |
| 7 | \( 1 + (0.831 - 0.555i)T \) |
| 11 | \( 1 + (-0.923 - 0.382i)T \) |
| 13 | \( 1 + (0.555 - 0.831i)T \) |
| 17 | \( 1 + (0.555 - 0.831i)T \) |
| 19 | \( 1 + (-0.831 - 0.555i)T \) |
| 23 | \( 1 + (0.195 + 0.980i)T \) |
| 29 | \( 1 + (-0.195 - 0.980i)T \) |
| 31 | \( 1 + (0.382 - 0.923i)T \) |
| 37 | \( 1 + (0.980 + 0.195i)T \) |
| 41 | \( 1 + (-0.980 + 0.195i)T \) |
| 43 | \( 1 + (0.707 - 0.707i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.923 - 0.382i)T \) |
| 59 | \( 1 + (-0.195 + 0.980i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.555 + 0.831i)T \) |
| 71 | \( 1 + (0.980 + 0.195i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 + (-0.382 + 0.923i)T \) |
| 83 | \( 1 + (-0.831 - 0.555i)T \) |
| 89 | \( 1 + (0.923 + 0.382i)T \) |
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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
−24.54733611783004837472470873696, −23.62859938595801010840630878221, −23.26282511707615368013690444605, −21.68727275643803166415651817500, −21.069072194413922793036288400270, −20.22030905761683309505185672171, −19.16287196503120534253392020990, −18.58874479343048418824933875545, −17.990990799265966615134747398644, −16.47829288433445523092691401782, −15.445471505057907819745448619688, −14.741706267046737085311245739911, −14.22039771538999201128539377850, −12.85088047256361711770459706547, −12.19209589528705685134287938241, −11.05284559115218782000184629336, −10.16960974816196827352154582689, −8.64370292031460561839245361183, −8.25428915561054820016230138521, −7.29292014676119189279669597538, −6.28717248098909115458376907591, −4.71759460169247527624802221587, −3.729384709130180744062169057047, −2.60722308680751722998369649881, −1.614206959136186843251075826241,
0.95356596339174961513737486517, 2.546005952474240190136019389281, 3.65452128027274543068007071475, 4.547170354239640060245622819040, 5.45972417222309976019938699558, 7.39593601095880844718045662685, 7.963988259707884573880678509036, 8.621244932278289175700319431246, 9.835133746635371110466472799523, 10.84796291664382378810489797921, 11.64116282587349477140256404118, 13.15343508817472374544785225865, 13.499631499581938857981396587779, 14.82463285035281755458794689024, 15.450392677332266578343325556477, 16.21822457582069366949936569809, 17.251913087708617379849658489266, 18.44892591319786604742553366903, 19.28148576525267965857547498472, 20.21967844591370632344312117790, 20.788779004702186046200375743911, 21.36332730136928391799115739115, 22.80430903368552600100497753629, 23.6593973575270654861926001488, 24.31892666078481970086992295071