| L(s) = 1 | − 3·2-s + 6·4-s − 5-s + 4·7-s − 10·8-s + 3·10-s + 7·11-s − 13-s − 12·14-s + 15·16-s + 6·17-s − 3·19-s − 6·20-s − 21·22-s − 2·23-s − 2·25-s + 3·26-s + 24·28-s − 29-s − 3·31-s − 21·32-s − 18·34-s − 4·35-s + 7·37-s + 9·38-s + 10·40-s − 4·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 0.447·5-s + 1.51·7-s − 3.53·8-s + 0.948·10-s + 2.11·11-s − 0.277·13-s − 3.20·14-s + 15/4·16-s + 1.45·17-s − 0.688·19-s − 1.34·20-s − 4.47·22-s − 0.417·23-s − 2/5·25-s + 0.588·26-s + 4.53·28-s − 0.185·29-s − 0.538·31-s − 3.71·32-s − 3.08·34-s − 0.676·35-s + 1.15·37-s + 1.45·38-s + 1.58·40-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 61^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 61^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578554098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578554098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 61 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + T + 3 T^{2} - 6 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 15 T^{3} + 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 43 T^{2} - 150 T^{3} + 43 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 33 T^{2} + 22 T^{3} + 33 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 188 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 56 T^{2} + 110 T^{3} + 56 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 31 T^{2} - 21 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 56 T^{2} + 56 T^{3} + 56 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 50 T^{2} + 194 T^{3} + 50 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 7 T + 46 T^{2} - 94 T^{3} + 46 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 467 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 113 T^{2} - 776 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 113 T^{2} - 544 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 11 T - 36 T^{2} - 1032 T^{3} - 36 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 23 T + 341 T^{2} - 3082 T^{3} + 341 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 21 T + 245 T^{2} - 2042 T^{3} + 245 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 27 T + 420 T^{2} + 4266 T^{3} + 420 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 299 T^{2} - 2763 T^{3} + 299 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 3 T + 129 T^{2} - 42 T^{3} + 129 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 11 T + 164 T^{2} - 1798 T^{3} + 164 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 191 T^{2} - 1892 T^{3} + 191 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 5 T + 284 T^{2} + 972 T^{3} + 284 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.867904544760637407189855569629, −8.338670361240084235768752539974, −8.248342970226689120037345636894, −8.163720946910673725159299379367, −7.57387246476818699505335348910, −7.49417606899587020296285317341, −7.46969977897348269598395783305, −6.88574375684352116241082193228, −6.55103099382117253862630666796, −6.52015093177128026316619384687, −5.96953127185242466684810529717, −5.63250762619963148069866584293, −5.59970030602896512363425022865, −4.99592738706006586395811204732, −4.48430178871476826379961281754, −4.37660737799863538870416477246, −3.82846540727988380696074467479, −3.47837216586679871804272213063, −3.41667926426076778842368630348, −2.50191690690808572462807579836, −2.23964545562122950471296015914, −1.89794029504344839682901547044, −1.46835823363898922655349767995, −0.892151096565085967413481317484, −0.73014257456017341271987376300,
0.73014257456017341271987376300, 0.892151096565085967413481317484, 1.46835823363898922655349767995, 1.89794029504344839682901547044, 2.23964545562122950471296015914, 2.50191690690808572462807579836, 3.41667926426076778842368630348, 3.47837216586679871804272213063, 3.82846540727988380696074467479, 4.37660737799863538870416477246, 4.48430178871476826379961281754, 4.99592738706006586395811204732, 5.59970030602896512363425022865, 5.63250762619963148069866584293, 5.96953127185242466684810529717, 6.52015093177128026316619384687, 6.55103099382117253862630666796, 6.88574375684352116241082193228, 7.46969977897348269598395783305, 7.49417606899587020296285317341, 7.57387246476818699505335348910, 8.163720946910673725159299379367, 8.248342970226689120037345636894, 8.338670361240084235768752539974, 8.867904544760637407189855569629
