| L(s) = 1 | + 3-s − 4·5-s − 7-s + 9-s − 2·11-s + 6·13-s − 4·15-s − 4·17-s + 4·19-s − 21-s + 2·23-s + 11·25-s + 27-s + 2·29-s − 2·33-s + 4·35-s − 2·37-s + 6·39-s + 4·43-s − 4·45-s + 12·47-s + 49-s − 4·51-s + 6·53-s + 8·55-s + 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 1.03·15-s − 0.970·17-s + 0.917·19-s − 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.192·27-s + 0.371·29-s − 0.348·33-s + 0.676·35-s − 0.328·37-s + 0.960·39-s + 0.609·43-s − 0.596·45-s + 1.75·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375268914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375268914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.365984762551205725158163743220, −8.617092129725433919332782114740, −8.106251167080634675502825469413, −7.30207867772318096349635238411, −6.59023808703041071787627483496, −5.30046762765077591142520164517, −4.12575993015631772148699876717, −3.64261842522368699265247670760, −2.68731442866574654307984003997, −0.837969571004769044543745728003,
0.837969571004769044543745728003, 2.68731442866574654307984003997, 3.64261842522368699265247670760, 4.12575993015631772148699876717, 5.30046762765077591142520164517, 6.59023808703041071787627483496, 7.30207867772318096349635238411, 8.106251167080634675502825469413, 8.617092129725433919332782114740, 9.365984762551205725158163743220
