| L(s) = 1 | − 3-s − 3·7-s + 9-s − 5·11-s − 13-s + 4·17-s − 3·19-s + 3·21-s − 23-s − 27-s − 3·29-s + 5·33-s − 2·37-s + 39-s − 7·41-s + 3·43-s + 4·47-s + 2·49-s − 4·51-s − 4·53-s + 3·57-s − 12·59-s − 6·61-s − 3·63-s − 12·67-s + 69-s + 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.970·17-s − 0.688·19-s + 0.654·21-s − 0.208·23-s − 0.192·27-s − 0.557·29-s + 0.870·33-s − 0.328·37-s + 0.160·39-s − 1.09·41-s + 0.457·43-s + 0.583·47-s + 2/7·49-s − 0.560·51-s − 0.549·53-s + 0.397·57-s − 1.56·59-s − 0.768·61-s − 0.377·63-s − 1.46·67-s + 0.120·69-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5485033546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5485033546\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.70736412087541943046274134302, −7.37990078668425542615709528450, −6.39506658334447590967085768969, −5.91290608379510874509002300392, −5.21040080955302151179466420993, −4.50150158467876115442340682788, −3.44579175556891513390285321177, −2.85982556991009903454508236888, −1.81210539844811414646194504746, −0.37126067759636396051179936859,
0.37126067759636396051179936859, 1.81210539844811414646194504746, 2.85982556991009903454508236888, 3.44579175556891513390285321177, 4.50150158467876115442340682788, 5.21040080955302151179466420993, 5.91290608379510874509002300392, 6.39506658334447590967085768969, 7.37990078668425542615709528450, 7.70736412087541943046274134302
