| L(s) = 1 | − 12·4-s + 84·16-s + 7·27-s − 448·64-s − 84·108-s − 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 78·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 6·4-s + 21·16-s + 1.34·27-s − 56·64-s − 8.08·108-s − 6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1791640781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1791640781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 7 T^{3} + p^{3} T^{6} \) |
| 59 | \( ( 1 + p T^{2} )^{3} \) |
| good | 2 | \( ( 1 + p T^{2} )^{6} \) |
| 5 | \( ( 1 - 21 T^{3} + p^{3} T^{6} )( 1 + 21 T^{3} + p^{3} T^{6} ) \) |
| 7 | \( ( 1 - 29 T^{3} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + p T^{2} )^{6} \) |
| 13 | \( ( 1 - p T^{2} )^{6} \) |
| 17 | \( ( 1 - 3 T + p T^{2} )^{3}( 1 + 3 T + p T^{2} )^{3} \) |
| 19 | \( ( 1 - 119 T^{3} + p^{3} T^{6} )^{2} \) |
| 23 | \( ( 1 + p T^{2} )^{6} \) |
| 29 | \( ( 1 - 159 T^{3} + p^{3} T^{6} )( 1 + 159 T^{3} + p^{3} T^{6} ) \) |
| 31 | \( ( 1 - p T^{2} )^{6} \) |
| 37 | \( ( 1 - p T^{2} )^{6} \) |
| 41 | \( ( 1 - 525 T^{3} + p^{3} T^{6} )( 1 + 525 T^{3} + p^{3} T^{6} ) \) |
| 43 | \( ( 1 - p T^{2} )^{6} \) |
| 47 | \( ( 1 + p T^{2} )^{6} \) |
| 53 | \( ( 1 - 327 T^{3} + p^{3} T^{6} )( 1 + 327 T^{3} + p^{3} T^{6} ) \) |
| 61 | \( ( 1 - p T^{2} )^{6} \) |
| 67 | \( ( 1 - p T^{2} )^{6} \) |
| 71 | \( ( 1 - 15 T + p T^{2} )^{3}( 1 + 15 T + p T^{2} )^{3} \) |
| 73 | \( ( 1 - p T^{2} )^{6} \) |
| 79 | \( ( 1 + 835 T^{3} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + p T^{2} )^{6} \) |
| 89 | \( ( 1 + p T^{2} )^{6} \) |
| 97 | \( ( 1 - p T^{2} )^{6} \) |
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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
−6.97986019234553109497116988072, −6.91357861522017802696229500212, −6.61989657582877123296151863307, −6.48410664301203497605651106638, −5.93485453385211538678088719703, −5.90702523545162679754096326291, −5.61371444067291297466970480936, −5.45614113122990597672285857486, −5.36430593724127608616428479917, −5.23223248435632000781719646647, −4.87970349241540486350579246227, −4.63005458357634827071880962273, −4.59663219353282400749579363017, −4.32469803135168192830013387671, −4.22510046070998564259599295211, −4.08726542167024137495585613196, −3.61760468280945731292185628031, −3.55441166804018318099222003724, −3.33928614684064914842433888603, −2.90372236907266968934588291955, −2.87618639114952677141802918645, −1.90759366497427453963034178453, −1.23454570450566170948994982900, −1.01085268182840106376343442750, −0.37985719913390069452166199875,
0.37985719913390069452166199875, 1.01085268182840106376343442750, 1.23454570450566170948994982900, 1.90759366497427453963034178453, 2.87618639114952677141802918645, 2.90372236907266968934588291955, 3.33928614684064914842433888603, 3.55441166804018318099222003724, 3.61760468280945731292185628031, 4.08726542167024137495585613196, 4.22510046070998564259599295211, 4.32469803135168192830013387671, 4.59663219353282400749579363017, 4.63005458357634827071880962273, 4.87970349241540486350579246227, 5.23223248435632000781719646647, 5.36430593724127608616428479917, 5.45614113122990597672285857486, 5.61371444067291297466970480936, 5.90702523545162679754096326291, 5.93485453385211538678088719703, 6.48410664301203497605651106638, 6.61989657582877123296151863307, 6.91357861522017802696229500212, 6.97986019234553109497116988072
Plot not available for L-functions of degree greater than 10.
