L(s) = 1 | + 2·2-s + 3·4-s + 6·5-s + 4·8-s + 12·10-s + 3·16-s + 16·19-s + 18·20-s + 4·23-s + 21·25-s − 12·29-s + 6·32-s + 32·38-s + 24·40-s − 16·43-s + 8·46-s + 8·47-s + 36·49-s + 42·50-s + 8·53-s − 24·58-s + 11·64-s − 48·71-s − 12·73-s + 48·76-s + 18·80-s − 32·86-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 2.68·5-s + 1.41·8-s + 3.79·10-s + 3/4·16-s + 3.67·19-s + 4.02·20-s + 0.834·23-s + 21/5·25-s − 2.22·29-s + 1.06·32-s + 5.19·38-s + 3.79·40-s − 2.43·43-s + 1.17·46-s + 1.16·47-s + 36/7·49-s + 5.93·50-s + 1.09·53-s − 3.15·58-s + 11/8·64-s − 5.69·71-s − 1.40·73-s + 5.50·76-s + 2.01·80-s − 3.45·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(17.57137012\) |
\(L(\frac12)\) |
\(\approx\) |
\(17.57137012\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T + T^{2} + p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - T )^{6} \) |
good | 7 | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{3} \) |
| 11 | \( 1 - 20 T^{2} + p T^{4} + 1944 T^{6} + p^{3} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 32 T^{2} + 747 T^{4} - 10688 T^{6} + 747 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 22 T^{2} + 895 T^{4} - 12052 T^{6} + 895 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 8 T + 53 T^{2} - 288 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( ( 1 - 2 T + 37 T^{2} - 60 T^{3} + 37 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 6 T + 59 T^{2} + 340 T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 106 T^{2} + 6495 T^{4} - 243052 T^{6} + 6495 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 144 T^{2} + 10235 T^{4} - 459424 T^{6} + 10235 p^{2} T^{8} - 144 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 112 T^{2} + 7987 T^{4} - 366304 T^{6} + 7987 p^{2} T^{8} - 112 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 8 T + 73 T^{2} + 752 T^{3} + 73 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( ( 1 - 4 T + 49 T^{2} - 24 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 4 T + 87 T^{2} - 72 T^{3} + 87 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 244 T^{2} + 26347 T^{4} - 1820968 T^{6} + 26347 p^{2} T^{8} - 244 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 182 T^{2} + 14439 T^{4} - 860564 T^{6} + 14439 p^{2} T^{8} - 182 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 161 T^{2} + 64 T^{3} + 161 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 24 T + 365 T^{2} + 3536 T^{3} + 365 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 6 T + 71 T^{2} + 52 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 138 T^{2} + 18623 T^{4} - 1324204 T^{6} + 18623 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 - 186 T^{2} + 19655 T^{4} - 1523596 T^{6} + 19655 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 416 T^{2} + 78163 T^{4} - 8732480 T^{6} + 78163 p^{2} T^{8} - 416 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 2 T + 87 T^{2} + 820 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−6.01772574654796768514878768774, −5.81391628355597829055086288547, −5.74371697477974943568139040056, −5.69575359372325674656708609283, −5.61343933811328148077967757650, −5.40372622183108962916038069424, −5.11713347030823826016667275002, −5.10978474655816132414590688472, −4.89967479406880544331053140005, −4.49266654208795169986007175079, −4.45452071375734487487687775055, −4.13864094642444116159495780480, −3.87592326277221873390286920757, −3.82187126917501070753920460209, −3.45853119481712164088158435787, −3.06637737534298201299556997526, −2.91436788598177751154732095300, −2.85290180223659006818558969714, −2.77504146160861910920654477091, −2.32439493219263104552664642614, −1.94424931743962889361801291061, −1.88731179255881144931631660115, −1.50692814017583657196135147188, −1.04825444583806461937419346741, −1.04488133756077640935245449536,
1.04488133756077640935245449536, 1.04825444583806461937419346741, 1.50692814017583657196135147188, 1.88731179255881144931631660115, 1.94424931743962889361801291061, 2.32439493219263104552664642614, 2.77504146160861910920654477091, 2.85290180223659006818558969714, 2.91436788598177751154732095300, 3.06637737534298201299556997526, 3.45853119481712164088158435787, 3.82187126917501070753920460209, 3.87592326277221873390286920757, 4.13864094642444116159495780480, 4.45452071375734487487687775055, 4.49266654208795169986007175079, 4.89967479406880544331053140005, 5.10978474655816132414590688472, 5.11713347030823826016667275002, 5.40372622183108962916038069424, 5.61343933811328148077967757650, 5.69575359372325674656708609283, 5.74371697477974943568139040056, 5.81391628355597829055086288547, 6.01772574654796768514878768774
Plot not available for L-functions of degree greater than 10.