| L(s) = 1 | − 4·4-s − 2·7-s + 10·13-s + 12·16-s − 14·19-s − 10·25-s + 8·28-s − 8·31-s + 22·37-s + 16·43-s − 11·49-s − 40·52-s − 2·61-s − 32·64-s + 10·67-s − 14·73-s + 56·76-s + 34·79-s − 20·91-s − 38·97-s + 40·100-s − 26·103-s + 4·109-s − 24·112-s − 22·121-s + 32·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.755·7-s + 2.77·13-s + 3·16-s − 3.21·19-s − 2·25-s + 1.51·28-s − 1.43·31-s + 3.61·37-s + 2.43·43-s − 1.57·49-s − 5.54·52-s − 0.256·61-s − 4·64-s + 1.22·67-s − 1.63·73-s + 6.42·76-s + 3.82·79-s − 2.09·91-s − 3.85·97-s + 4·100-s − 2.56·103-s + 0.383·109-s − 2.26·112-s − 2·121-s + 2.87·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3467791637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3467791637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.00705820570875902116997986320, −19.50763388930887128039872729916, −19.50763388930887128039872729916, −18.55924110538682425020987188449, −18.55924110538682425020987188449, −17.54958210256512249523818407718, −17.54958210256512249523818407718, −16.30001713525188163931408346208, −16.30001713525188163931408346208, −14.89211076740924954992271086519, −14.89211076740924954992271086519, −13.58211369833951574831601484907, −13.58211369833951574831601484907, −12.71563949069013251763955822657, −12.71563949069013251763955822657, −10.90872829298908742564232271224, −10.90872829298908742564232271224, −9.429199208210393651255146412199, −9.429199208210393651255146412199, −8.217650367462526737991465554229, −8.217650367462526737991465554229, −6.04893540000987436402649531985, −6.04893540000987436402649531985, −4.04304401379743272242521882180, −4.04304401379743272242521882180,
4.04304401379743272242521882180, 4.04304401379743272242521882180, 6.04893540000987436402649531985, 6.04893540000987436402649531985, 8.217650367462526737991465554229, 8.217650367462526737991465554229, 9.429199208210393651255146412199, 9.429199208210393651255146412199, 10.90872829298908742564232271224, 10.90872829298908742564232271224, 12.71563949069013251763955822657, 12.71563949069013251763955822657, 13.58211369833951574831601484907, 13.58211369833951574831601484907, 14.89211076740924954992271086519, 14.89211076740924954992271086519, 16.30001713525188163931408346208, 16.30001713525188163931408346208, 17.54958210256512249523818407718, 17.54958210256512249523818407718, 18.55924110538682425020987188449, 18.55924110538682425020987188449, 19.50763388930887128039872729916, 19.50763388930887128039872729916, 21.00705820570875902116997986320
