| L(s) = 1 | + 3-s + 3·5-s + 3·7-s + 9-s − 11-s + 2·13-s + 3·15-s − 5·17-s + 19-s + 3·21-s + 4·23-s + 4·25-s + 27-s + 6·29-s + 2·31-s − 33-s + 9·35-s − 8·37-s + 2·39-s − 8·41-s + 13·43-s + 3·45-s − 13·47-s + 2·49-s − 5·51-s + 6·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.774·15-s − 1.21·17-s + 0.229·19-s + 0.654·21-s + 0.834·23-s + 4/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.174·33-s + 1.52·35-s − 1.31·37-s + 0.320·39-s − 1.24·41-s + 1.98·43-s + 0.447·45-s − 1.89·47-s + 2/7·49-s − 0.700·51-s + 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.646559279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.646559279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.565568506209819562343972632484, −8.041217035345116672180987625593, −6.92914629564428708521802369462, −6.44874475425758117572184164743, −5.32884476536624310380635764204, −4.94014466378124396829278693187, −3.90678377596813653368860851375, −2.72661795833497144662962533122, −2.03870573124656232297141280260, −1.21263824189784456383522880818,
1.21263824189784456383522880818, 2.03870573124656232297141280260, 2.72661795833497144662962533122, 3.90678377596813653368860851375, 4.94014466378124396829278693187, 5.32884476536624310380635764204, 6.44874475425758117572184164743, 6.92914629564428708521802369462, 8.041217035345116672180987625593, 8.565568506209819562343972632484
