| L(s) = 1 | + 6·3-s + 4-s − 2·8-s + 21·9-s + 6·12-s + 3·16-s − 12·24-s + 56·27-s − 12·31-s − 4·32-s + 21·36-s + 8·37-s − 20·41-s − 8·43-s + 18·48-s + 6·49-s + 12·53-s + 64-s + 24·67-s − 8·71-s − 42·72-s − 36·79-s + 126·81-s + 32·83-s + 28·89-s − 72·93-s − 24·96-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 1/2·4-s − 0.707·8-s + 7·9-s + 1.73·12-s + 3/4·16-s − 2.44·24-s + 10.7·27-s − 2.15·31-s − 0.707·32-s + 7/2·36-s + 1.31·37-s − 3.12·41-s − 1.21·43-s + 2.59·48-s + 6/7·49-s + 1.64·53-s + 1/8·64-s + 2.93·67-s − 0.949·71-s − 4.94·72-s − 4.05·79-s + 14·81-s + 3.51·83-s + 2.96·89-s − 7.46·93-s − 2.44·96-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.33816588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.33816588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{2} + p T^{3} - p T^{4} + p^{3} T^{6} \) |
| 3 | \( ( 1 - T )^{6} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 11 T^{2} + 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 + 6 T + 77 T^{2} + 308 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 4 T + 51 T^{2} - 40 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 4 T + 65 T^{2} + 216 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{6} \) |
| 71 | \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 16 T + 265 T^{2} - 2400 T^{3} + 265 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.85474671745086958442085103907, −5.56166351968383264339307476619, −5.18402161556748117293998851341, −5.08069673183854232950890688261, −5.01823876768522536197359845525, −4.88130287791933523267089856338, −4.75519815235280460863989841102, −4.31585325601172174806390557034, −3.94944728548798118360630720725, −3.94742907484286548935946397024, −3.91080918901735059205203971493, −3.64469976760805615710973763927, −3.49765089720432791333809877007, −3.28471518732951362324936581607, −3.23804994450226999943665244299, −2.86510314507904884962965152604, −2.67846506628785444712207839419, −2.55400430444505181065966949595, −2.45652533239228763939897541453, −1.88669363849307231964633850692, −1.78370358698595718345923454787, −1.70706145026574424906716649019, −1.69841650150167190005975056670, −0.833535042035113445271247040078, −0.64593008690498888212488933530,
0.64593008690498888212488933530, 0.833535042035113445271247040078, 1.69841650150167190005975056670, 1.70706145026574424906716649019, 1.78370358698595718345923454787, 1.88669363849307231964633850692, 2.45652533239228763939897541453, 2.55400430444505181065966949595, 2.67846506628785444712207839419, 2.86510314507904884962965152604, 3.23804994450226999943665244299, 3.28471518732951362324936581607, 3.49765089720432791333809877007, 3.64469976760805615710973763927, 3.91080918901735059205203971493, 3.94742907484286548935946397024, 3.94944728548798118360630720725, 4.31585325601172174806390557034, 4.75519815235280460863989841102, 4.88130287791933523267089856338, 5.01823876768522536197359845525, 5.08069673183854232950890688261, 5.18402161556748117293998851341, 5.56166351968383264339307476619, 5.85474671745086958442085103907
Plot not available for L-functions of degree greater than 10.
