L(s) = 1 | + 3·3-s + 4·4-s + 9·9-s + 12·12-s − 23·13-s + 16·16-s − 11·19-s + 25·25-s + 27·27-s + 13·31-s + 36·36-s − 73·37-s − 69·39-s − 61·43-s + 48·48-s − 92·52-s − 33·57-s − 74·61-s + 64·64-s − 13·67-s + 97·73-s + 75·75-s − 44·76-s + 11·79-s + 81·81-s + 39·93-s − 2·97-s + ⋯ |
L(s) = 1 | + 3-s + 4-s + 9-s + 12-s − 1.76·13-s + 16-s − 0.578·19-s + 25-s + 27-s + 0.419·31-s + 36-s − 1.97·37-s − 1.76·39-s − 1.41·43-s + 48-s − 1.76·52-s − 0.578·57-s − 1.21·61-s + 64-s − 0.194·67-s + 1.32·73-s + 75-s − 0.578·76-s + 0.139·79-s + 81-s + 0.419·93-s − 0.0206·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.281132653\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.281132653\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 13 T + p^{2} T^{2} \) |
| 37 | \( 1 + 73 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 61 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 74 T + p^{2} T^{2} \) |
| 67 | \( 1 + 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 97 T + p^{2} T^{2} \) |
| 79 | \( 1 - 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 2 T + p^{2} T^{2} \) |
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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
−12.68335225490082405301353772104, −12.02495324137181980318558862894, −10.61475239491124312656255385715, −9.824413471321259790039875562075, −8.560827121366394777761342863881, −7.45783461989240393020086773671, −6.69878656362012781762069178695, −4.89314783644220342530353574901, −3.17415659105302014468945646645, −2.03103217404730035474201708510,
2.03103217404730035474201708510, 3.17415659105302014468945646645, 4.89314783644220342530353574901, 6.69878656362012781762069178695, 7.45783461989240393020086773671, 8.560827121366394777761342863881, 9.824413471321259790039875562075, 10.61475239491124312656255385715, 12.02495324137181980318558862894, 12.68335225490082405301353772104