L(s) = 1 | + 3·11-s + 27-s − 3·41-s + 3·43-s − 6·59-s + 3·67-s + 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 3·11-s + 27-s − 3·41-s + 3·43-s − 6·59-s + 3·67-s + 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9324120623\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9324120623\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T^{3} + T^{6} \) |
good | 5 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 7 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 11 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 13 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 17 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 19 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 23 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 29 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 31 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 41 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
| 43 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 47 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 53 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 59 | \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \) |
| 61 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 67 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 71 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 73 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 79 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 83 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 89 | \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \) |
| 97 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.82120811786269020488655950824, −5.27353005254842872102181965405, −5.25725246553453368970832890388, −5.12943223799191162585901759655, −4.96465952706016470716110972001, −4.83887828113249312189562565079, −4.77382009248142497053692253420, −4.29690578283899426179963838290, −4.25728877682293028888567227115, −4.14925594983012218675113337399, −3.97024591520817943054651417412, −3.81160705595061180820198190542, −3.48038191394335792638921780288, −3.47266272589055496995336154381, −3.41635591593603092218720761330, −2.95271955397086025546705325804, −2.80530936422477318736258737986, −2.58004789456972379620689128136, −2.54837736362177468293033411053, −2.01109970149798648382494256556, −1.89445748414108061807889473431, −1.60065576960816549234678278242, −1.30184874323710865942771920751, −1.27439171901337151242175599893, −0.923867651047348099398639848761,
0.923867651047348099398639848761, 1.27439171901337151242175599893, 1.30184874323710865942771920751, 1.60065576960816549234678278242, 1.89445748414108061807889473431, 2.01109970149798648382494256556, 2.54837736362177468293033411053, 2.58004789456972379620689128136, 2.80530936422477318736258737986, 2.95271955397086025546705325804, 3.41635591593603092218720761330, 3.47266272589055496995336154381, 3.48038191394335792638921780288, 3.81160705595061180820198190542, 3.97024591520817943054651417412, 4.14925594983012218675113337399, 4.25728877682293028888567227115, 4.29690578283899426179963838290, 4.77382009248142497053692253420, 4.83887828113249312189562565079, 4.96465952706016470716110972001, 5.12943223799191162585901759655, 5.25725246553453368970832890388, 5.27353005254842872102181965405, 5.82120811786269020488655950824
Plot not available for L-functions of degree greater than 10.