| L(s) = 1 | + 3-s − 3·7-s + 9-s − 3·11-s − 17-s + 7·19-s − 3·21-s + 4·23-s + 27-s + 29-s + 10·31-s − 3·33-s + 37-s − 3·41-s − 10·43-s + 3·47-s + 2·49-s − 51-s − 9·53-s + 7·57-s + 10·59-s − 12·61-s − 3·63-s + 10·67-s + 4·69-s − 13·73-s + 9·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.242·17-s + 1.60·19-s − 0.654·21-s + 0.834·23-s + 0.192·27-s + 0.185·29-s + 1.79·31-s − 0.522·33-s + 0.164·37-s − 0.468·41-s − 1.52·43-s + 0.437·47-s + 2/7·49-s − 0.140·51-s − 1.23·53-s + 0.927·57-s + 1.30·59-s − 1.53·61-s − 0.377·63-s + 1.22·67-s + 0.481·69-s − 1.52·73-s + 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.055793614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055793614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.58853527319409, −15.29924895239724, −14.53116815933340, −13.82101503462610, −13.49315274718983, −13.04258795468094, −12.47933952475535, −11.83168459995481, −11.29149683579348, −10.45080059638628, −9.960064256143718, −9.627318101015784, −8.954176622252083, −8.298835197148789, −7.783889913377972, −7.068190577649401, −6.624585864206645, −5.889922366753494, −5.119541355270877, −4.604903418455253, −3.606705512857884, −2.982270131464028, −2.732056956770397, −1.554151940525186, −0.5742687454176715,
0.5742687454176715, 1.554151940525186, 2.732056956770397, 2.982270131464028, 3.606705512857884, 4.604903418455253, 5.119541355270877, 5.889922366753494, 6.624585864206645, 7.068190577649401, 7.783889913377972, 8.298835197148789, 8.954176622252083, 9.627318101015784, 9.960064256143718, 10.45080059638628, 11.29149683579348, 11.83168459995481, 12.47933952475535, 13.04258795468094, 13.49315274718983, 13.82101503462610, 14.53116815933340, 15.29924895239724, 15.58853527319409
