| L(s) = 1 | − 3-s + 5-s − 2·7-s − 2·9-s − 13-s − 15-s − 4·17-s + 4·19-s + 2·21-s + 23-s + 25-s + 5·27-s + 3·29-s − 31-s − 2·35-s + 8·37-s + 39-s − 5·41-s + 6·43-s − 2·45-s + 9·47-s − 3·49-s + 4·51-s − 2·53-s − 4·57-s + 4·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s − 0.277·13-s − 0.258·15-s − 0.970·17-s + 0.917·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.179·31-s − 0.338·35-s + 1.31·37-s + 0.160·39-s − 0.780·41-s + 0.914·43-s − 0.298·45-s + 1.31·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s − 0.529·57-s + 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.42065049241711421621899415555, −6.69163029015836848593168981812, −6.17607206171084367464022956748, −5.52658163134853430633232000366, −4.87737397782985134573311743465, −3.99457630215215917188025143286, −2.96122768826948736543671512093, −2.44437109725078375286149217229, −1.09943289091794875293665468274, 0,
1.09943289091794875293665468274, 2.44437109725078375286149217229, 2.96122768826948736543671512093, 3.99457630215215917188025143286, 4.87737397782985134573311743465, 5.52658163134853430633232000366, 6.17607206171084367464022956748, 6.69163029015836848593168981812, 7.42065049241711421621899415555
