| L(s) = 1 | + 12·7-s − 8-s − 9·11-s − 8·27-s + 6·31-s + 60·37-s + 69·49-s − 12·56-s − 108·77-s − 9·83-s + 9·88-s + 6·103-s − 90·113-s + 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 96·189-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 0.353·8-s − 2.71·11-s − 1.53·27-s + 1.07·31-s + 9.86·37-s + 69/7·49-s − 1.60·56-s − 12.3·77-s − 0.987·83-s + 0.959·88-s + 0.591·103-s − 8.46·113-s + 5.45·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 6.98·189-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.946671551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.946671551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 8 T^{3} + 37 T^{6} + 8 p^{3} T^{9} + p^{6} T^{12} \) |
| 5 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 5 T + p T^{2} )^{3}( 1 + T + p T^{2} )^{3} \) |
| 11 | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{3} \) |
| 13 | \( ( 1 - 19 T^{3} + p^{3} T^{6} )( 1 + 89 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( 1 + 90 T^{3} + 3187 T^{6} + 90 p^{3} T^{9} + p^{6} T^{12} \) |
| 23 | \( 1 + 198 T^{3} + 27037 T^{6} + 198 p^{3} T^{9} + p^{6} T^{12} \) |
| 29 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{3} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{6} \) |
| 41 | \( 1 + 378 T^{3} + 73963 T^{6} + 378 p^{3} T^{9} + p^{6} T^{12} \) |
| 43 | \( 1 + 452 T^{3} + 124797 T^{6} + 452 p^{3} T^{9} + p^{6} T^{12} \) |
| 47 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( 1 + 738 T^{3} + 395767 T^{6} + 738 p^{3} T^{9} + p^{6} T^{12} \) |
| 59 | \( 1 - 864 T^{3} + 541117 T^{6} - 864 p^{3} T^{9} + p^{6} T^{12} \) |
| 61 | \( 1 + 668 T^{3} + 219243 T^{6} + 668 p^{3} T^{9} + p^{6} T^{12} \) |
| 67 | \( 1 - 1064 T^{3} + 831333 T^{6} - 1064 p^{3} T^{9} + p^{6} T^{12} \) |
| 71 | \( 1 - 1062 T^{3} + 769933 T^{6} - 1062 p^{3} T^{9} + p^{6} T^{12} \) |
| 73 | \( 1 + 218 T^{3} - 341493 T^{6} + 218 p^{3} T^{9} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 1387 T^{3} + p^{3} T^{6} )( 1 + 503 T^{3} + p^{3} T^{6} ) \) |
| 83 | \( ( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{3} \) |
| 89 | \( 1 + 1386 T^{3} + 1216027 T^{6} + 1386 p^{3} T^{9} + p^{6} T^{12} \) |
| 97 | \( 1 + 34 T^{3} - 911517 T^{6} + 34 p^{3} T^{9} + p^{6} T^{12} \) |
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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
−5.61510673657310915259443832053, −5.30858373058272081872335747447, −5.13496683457295790015460276343, −5.13451626710826293614325397983, −4.87377015265457547743111652272, −4.54248725672155472810948125122, −4.48357259861254960198815730149, −4.45284417957114147892192334220, −4.32562427520217002642406007901, −4.32101285497811208830324303375, −3.94931704941468278148591978383, −3.73794623536744336668601287618, −3.49637664999821708946230897709, −2.86761895797600401730088869190, −2.78987004514008445659399093857, −2.76386385495326814592204699271, −2.58811909730549232157672224177, −2.52679004143879637951615588848, −2.21075371326934445856806379838, −1.82069310926161430951321698942, −1.66196881199878503663444249729, −1.55747643481342866717932167875, −1.01083604005651643567577493704, −0.806479531978887987808804451369, −0.60927823552172014756177828807,
0.60927823552172014756177828807, 0.806479531978887987808804451369, 1.01083604005651643567577493704, 1.55747643481342866717932167875, 1.66196881199878503663444249729, 1.82069310926161430951321698942, 2.21075371326934445856806379838, 2.52679004143879637951615588848, 2.58811909730549232157672224177, 2.76386385495326814592204699271, 2.78987004514008445659399093857, 2.86761895797600401730088869190, 3.49637664999821708946230897709, 3.73794623536744336668601287618, 3.94931704941468278148591978383, 4.32101285497811208830324303375, 4.32562427520217002642406007901, 4.45284417957114147892192334220, 4.48357259861254960198815730149, 4.54248725672155472810948125122, 4.87377015265457547743111652272, 5.13451626710826293614325397983, 5.13496683457295790015460276343, 5.30858373058272081872335747447, 5.61510673657310915259443832053
Plot not available for L-functions of degree greater than 10.
