| L(s) = 1 | + 6·7-s + 8·13-s + 4·19-s − 9·25-s − 14·31-s − 12·37-s − 4·43-s + 13·49-s − 16·61-s − 20·67-s + 2·73-s + 32·79-s + 48·91-s − 2·97-s + 8·103-s + 20·109-s + 3·121-s + 127-s + 131-s + 24·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 2.21·13-s + 0.917·19-s − 9/5·25-s − 2.51·31-s − 1.97·37-s − 0.609·43-s + 13/7·49-s − 2.04·61-s − 2.44·67-s + 0.234·73-s + 3.60·79-s + 5.03·91-s − 0.203·97-s + 0.788·103-s + 1.91·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 2.08·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.820048987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820048987\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.49034723509072962140764470290, −9.460094847454279126868063019693, −9.046545135813963295999252728111, −8.554704477910262490194224105907, −8.092105179993141599508863549634, −7.62368049380914397426496743518, −7.25448367711369655898353657421, −6.23138020629374792100300833919, −5.79729187121046804129336776899, −5.18432945826177342955590271236, −4.71525232702415007486392238194, −3.74514947909231141400939990651, −3.48503979598989868638885321849, −1.73582051333148490489274132763, −1.64081961055094255152301138209,
1.64081961055094255152301138209, 1.73582051333148490489274132763, 3.48503979598989868638885321849, 3.74514947909231141400939990651, 4.71525232702415007486392238194, 5.18432945826177342955590271236, 5.79729187121046804129336776899, 6.23138020629374792100300833919, 7.25448367711369655898353657421, 7.62368049380914397426496743518, 8.092105179993141599508863549634, 8.554704477910262490194224105907, 9.046545135813963295999252728111, 9.460094847454279126868063019693, 10.49034723509072962140764470290
