| L(s) = 1 | + 5·5-s + 22·7-s − 9·11-s + 17·13-s + 75·17-s + 4·19-s + 183·23-s + 25·25-s − 129·29-s + 187·31-s + 110·35-s − 34·37-s − 264·41-s − 443·43-s + 609·47-s + 141·49-s + 228·53-s − 45·55-s + 60·59-s − 454·61-s + 85·65-s + 244·67-s + 444·71-s + 398·73-s − 198·77-s + 349·79-s + 1.03e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.18·7-s − 0.246·11-s + 0.362·13-s + 1.07·17-s + 0.0482·19-s + 1.65·23-s + 1/5·25-s − 0.826·29-s + 1.08·31-s + 0.531·35-s − 0.151·37-s − 1.00·41-s − 1.57·43-s + 1.89·47-s + 0.411·49-s + 0.590·53-s − 0.110·55-s + 0.132·59-s − 0.952·61-s + 0.162·65-s + 0.444·67-s + 0.742·71-s + 0.638·73-s − 0.293·77-s + 0.497·79-s + 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.330546112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.330546112\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 17 T + p^{3} T^{2} \) |
| 17 | \( 1 - 75 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 183 T + p^{3} T^{2} \) |
| 29 | \( 1 + 129 T + p^{3} T^{2} \) |
| 31 | \( 1 - 187 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 264 T + p^{3} T^{2} \) |
| 43 | \( 1 + 443 T + p^{3} T^{2} \) |
| 47 | \( 1 - 609 T + p^{3} T^{2} \) |
| 53 | \( 1 - 228 T + p^{3} T^{2} \) |
| 59 | \( 1 - 60 T + p^{3} T^{2} \) |
| 61 | \( 1 + 454 T + p^{3} T^{2} \) |
| 67 | \( 1 - 244 T + p^{3} T^{2} \) |
| 71 | \( 1 - 444 T + p^{3} T^{2} \) |
| 73 | \( 1 - 398 T + p^{3} T^{2} \) |
| 79 | \( 1 - 349 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 89 | \( 1 + 852 T + p^{3} T^{2} \) |
| 97 | \( 1 - 914 T + p^{3} T^{2} \) |
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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
−8.654202316821057140918092628489, −7.999058309430798488234510332224, −7.25219277070444864511342378745, −6.36577497883118996761731118850, −5.28938706969638923975383514911, −5.01042023983484593860742106815, −3.79588240037927568904168295772, −2.80024011847704546839324045486, −1.70102749899987403393942908704, −0.891014188946412056084724941756,
0.891014188946412056084724941756, 1.70102749899987403393942908704, 2.80024011847704546839324045486, 3.79588240037927568904168295772, 5.01042023983484593860742106815, 5.28938706969638923975383514911, 6.36577497883118996761731118850, 7.25219277070444864511342378745, 7.999058309430798488234510332224, 8.654202316821057140918092628489
