| L(s) = 1 | − 7-s + 3·11-s − 4·17-s − 2·19-s − 23-s − 5·25-s − 4·29-s − 9·31-s − 3·37-s − 5·41-s + 4·43-s + 9·47-s + 49-s + 4·53-s + 10·59-s + 5·61-s − 11·67-s − 16·71-s − 11·73-s − 3·77-s − 5·79-s − 4·83-s − 2·89-s + 7·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s − 0.970·17-s − 0.458·19-s − 0.208·23-s − 25-s − 0.742·29-s − 1.61·31-s − 0.493·37-s − 0.780·41-s + 0.609·43-s + 1.31·47-s + 1/7·49-s + 0.549·53-s + 1.30·59-s + 0.640·61-s − 1.34·67-s − 1.89·71-s − 1.28·73-s − 0.341·77-s − 0.562·79-s − 0.439·83-s − 0.211·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9006350432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9006350432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.88026171013399, −13.38032897279120, −12.99922941232443, −12.50288365546342, −11.83426057025855, −11.53523254199470, −11.03184621022807, −10.28216813587110, −10.11263680838916, −9.210840073658888, −8.950705461243084, −8.633749844146254, −7.721766624523182, −7.224576584882995, −6.888339220099542, −6.111617281318475, −5.804767948945644, −5.177964753176024, −4.248747188306679, −4.027839262768933, −3.446649593533199, −2.595017558535467, −1.968892620875861, −1.411545434655068, −0.2968421810427553,
0.2968421810427553, 1.411545434655068, 1.968892620875861, 2.595017558535467, 3.446649593533199, 4.027839262768933, 4.248747188306679, 5.177964753176024, 5.804767948945644, 6.111617281318475, 6.888339220099542, 7.224576584882995, 7.721766624523182, 8.633749844146254, 8.950705461243084, 9.210840073658888, 10.11263680838916, 10.28216813587110, 11.03184621022807, 11.53523254199470, 11.83426057025855, 12.50288365546342, 12.99922941232443, 13.38032897279120, 13.88026171013399
