| L(s) = 1 | + 3-s + 7-s + 9-s + 11-s − 4·13-s − 2·17-s + 4·19-s + 21-s + 2·23-s + 27-s + 2·31-s + 33-s + 8·37-s − 4·39-s − 6·41-s + 4·43-s + 49-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s − 6·61-s + 63-s + 4·67-s + 2·69-s − 4·73-s + 77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 0.417·23-s + 0.192·27-s + 0.359·31-s + 0.174·33-s + 1.31·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.240·69-s − 0.468·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.884203619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.884203619\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.52189363846295, −11.94925345506615, −11.75935422495317, −11.09846373292906, −10.76521254884490, −10.11799352123184, −9.632369142015169, −9.478626806536643, −8.829123546144333, −8.422912910022086, −7.918274623211606, −7.463310719642290, −7.059878808301439, −6.657601025822708, −5.948590190864068, −5.473509684156486, −4.869818864999878, −4.528356624051130, −4.003864706112105, −3.347969796389265, −2.849027166901404, −2.346800036012836, −1.833647445386811, −1.096300849931958, −0.5224764355646214,
0.5224764355646214, 1.096300849931958, 1.833647445386811, 2.346800036012836, 2.849027166901404, 3.347969796389265, 4.003864706112105, 4.528356624051130, 4.869818864999878, 5.473509684156486, 5.948590190864068, 6.657601025822708, 7.059878808301439, 7.463310719642290, 7.918274623211606, 8.422912910022086, 8.829123546144333, 9.478626806536643, 9.632369142015169, 10.11799352123184, 10.76521254884490, 11.09846373292906, 11.75935422495317, 11.94925345506615, 12.52189363846295
