| L(s) = 1 | − 6·7-s + 2·8-s − 6·11-s + 6·23-s + 6·43-s + 21·49-s − 12·56-s + 64-s + 36·77-s − 12·88-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 36·161-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 12·184-s + 191-s + ⋯ |
| L(s) = 1 | − 6·7-s + 2·8-s − 6·11-s + 6·23-s + 6·43-s + 21·49-s − 12·56-s + 64-s + 36·77-s − 12·88-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 36·161-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 12·184-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5825329066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5825329066\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( ( 1 + T )^{6} \) |
| 11 | \( ( 1 + T )^{6} \) |
| 43 | \( ( 1 - T )^{6} \) |
| good | 2 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 3 | \( 1 - T^{6} + T^{12} \) |
| 5 | \( 1 - T^{6} + T^{12} \) |
| 13 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 17 | \( 1 - T^{6} + T^{12} \) |
| 19 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 23 | \( ( 1 - T + T^{2} )^{6} \) |
| 29 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 31 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 37 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 41 | \( 1 - T^{6} + T^{12} \) |
| 47 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 53 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 59 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 61 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 67 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 71 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 73 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 79 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 83 | \( 1 - T^{6} + T^{12} \) |
| 89 | \( ( 1 + T^{2} )^{6} \) |
| 97 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
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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
−4.62845025381119585935137093097, −4.56807208083852890955816004050, −4.30592251167204352831845041925, −4.24918905050462287344085091642, −4.22750218161341761354355288077, −3.89051929061376608274061306591, −3.73364855203391378193108226767, −3.36696554176594540161337435032, −3.36114866313686682902960155952, −3.12834159359762304708597431312, −3.11268737081187728429117616230, −3.07792495138543732068890134561, −2.96619327250389796491487991684, −2.77607906931930552135157304236, −2.63768066237565615846388638570, −2.47600462797563496725773151650, −2.32042870900223378047411027926, −2.28007286410743235673800277247, −2.16954680516121578815263408390, −1.76626982298360311091601119003, −1.15314755041358650459711425266, −0.981465568049694033529470700852, −0.864784803723543020446377103744, −0.58239319241444963534525937350, −0.45422024122663926346917650619,
0.45422024122663926346917650619, 0.58239319241444963534525937350, 0.864784803723543020446377103744, 0.981465568049694033529470700852, 1.15314755041358650459711425266, 1.76626982298360311091601119003, 2.16954680516121578815263408390, 2.28007286410743235673800277247, 2.32042870900223378047411027926, 2.47600462797563496725773151650, 2.63768066237565615846388638570, 2.77607906931930552135157304236, 2.96619327250389796491487991684, 3.07792495138543732068890134561, 3.11268737081187728429117616230, 3.12834159359762304708597431312, 3.36114866313686682902960155952, 3.36696554176594540161337435032, 3.73364855203391378193108226767, 3.89051929061376608274061306591, 4.22750218161341761354355288077, 4.24918905050462287344085091642, 4.30592251167204352831845041925, 4.56807208083852890955816004050, 4.62845025381119585935137093097
Plot not available for L-functions of degree greater than 10.
