| L(s) = 1 | − 729·3-s − 2.04e3·4-s − 7.71e4·7-s + 3.54e5·9-s + 1.49e6·12-s + 2.52e7·19-s + 5.62e7·21-s + 4.88e7·25-s − 1.29e8·27-s + 1.58e8·28-s + 2.02e8·31-s − 7.25e8·36-s + 6.63e8·37-s − 3.53e9·43-s + 3.97e9·49-s − 1.83e10·57-s + 2.15e10·61-s − 2.73e10·63-s + 8.58e9·64-s + 2.14e10·67-s + 4.27e9·73-s − 3.55e10·75-s − 5.16e10·76-s − 2.14e10·79-s + 3.13e10·81-s − 1.15e11·84-s − 1.47e11·93-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s − 1.73·7-s + 2·9-s + 1.73·12-s + 2.33·19-s + 3.00·21-s + 25-s − 1.73·27-s + 1.73·28-s + 1.27·31-s − 2·36-s + 1.57·37-s − 3.66·43-s + 2.01·49-s − 4.05·57-s + 3.26·61-s − 3.47·63-s + 64-s + 1.93·67-s + 0.241·73-s − 1.73·75-s − 2.33·76-s − 0.782·79-s + 81-s − 3.00·84-s − 2.20·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.6253944317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6253944317\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T + p^{11} T^{2} \) |
| 7 | $C_2$ | \( 1 + 77153 T + p^{11} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p^{11} T^{2} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{11} T^{2} + p^{22} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2382989 T + p^{11} T^{2} )( 1 + 2382989 T + p^{11} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 20581901 T + p^{11} T^{2} )( 1 - 4655368 T + p^{11} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{11} T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 249734764 T + p^{11} T^{2} )( 1 + 46742179 T + p^{11} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 782919730 T + p^{11} T^{2} )( 1 + 119374607 T + p^{11} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 1768358135 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{11} T^{2} + p^{22} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12977292913 T + p^{11} T^{2} )( 1 - 8546352539 T + p^{11} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 15458751248 T + p^{11} T^{2} )( 1 - 5951291615 T + p^{11} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 19805520230 T + p^{11} T^{2} )( 1 + 15533161001 T + p^{11} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 32885832404 T + p^{11} T^{2} )( 1 + 54296224537 T + p^{11} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 112637211442 T + p^{11} T^{2} )( 1 + 112637211442 T + p^{11} T^{2} ) \) |
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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
−15.87120685711821636619520165145, −15.70239018114830833869061258745, −14.48745541171504812652862850540, −13.54625619543939111656750465412, −13.14209106355966646318866666612, −12.66519505913501434828279708948, −11.68185929555701624873536147365, −11.52488076636173629532336542752, −10.17615987994714299937514811430, −9.886125155080246848289007398095, −9.354901834655399639726450017452, −8.183724634892179577670966637019, −6.76196309098378319371975946692, −6.69072964154620511027888855633, −5.44999995668248908107810621960, −5.05403879296783702763291689479, −3.95667666988824479989651290073, −2.99152536230134208411209198025, −0.984079518132159641243900442276, −0.47020608590198100993697020433,
0.47020608590198100993697020433, 0.984079518132159641243900442276, 2.99152536230134208411209198025, 3.95667666988824479989651290073, 5.05403879296783702763291689479, 5.44999995668248908107810621960, 6.69072964154620511027888855633, 6.76196309098378319371975946692, 8.183724634892179577670966637019, 9.354901834655399639726450017452, 9.886125155080246848289007398095, 10.17615987994714299937514811430, 11.52488076636173629532336542752, 11.68185929555701624873536147365, 12.66519505913501434828279708948, 13.14209106355966646318866666612, 13.54625619543939111656750465412, 14.48745541171504812652862850540, 15.70239018114830833869061258745, 15.87120685711821636619520165145
