| L(s) = 1 | + 3·5-s + 7-s + 3·11-s + 13-s − 12·17-s − 8·19-s − 3·23-s + 5·25-s + 3·29-s − 5·31-s + 3·35-s + 4·37-s + 3·41-s + 43-s − 9·47-s + 7·49-s + 12·53-s + 9·55-s − 3·59-s + 13·61-s + 3·65-s + 7·67-s + 24·71-s − 20·73-s + 3·77-s − 11·79-s − 9·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.904·11-s + 0.277·13-s − 2.91·17-s − 1.83·19-s − 0.625·23-s + 25-s + 0.557·29-s − 0.898·31-s + 0.507·35-s + 0.657·37-s + 0.468·41-s + 0.152·43-s − 1.31·47-s + 49-s + 1.64·53-s + 1.21·55-s − 0.390·59-s + 1.66·61-s + 0.372·65-s + 0.855·67-s + 2.84·71-s − 2.34·73-s + 0.341·77-s − 1.23·79-s − 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243642255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243642255\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−13.91878415200316710239896047481, −13.40695411814678904441797055855, −12.94451275136071770986430709731, −12.75603729900143268682568947074, −11.73487884299972956846533319769, −11.30716766037086820526144366698, −10.79424484238732835664106339963, −10.37590059688490989951133950811, −9.596763812420851867230064412340, −9.166064641133274764946146751231, −8.535061726195564870590584830362, −8.353545475517416957628422948266, −6.85060009229583151514173711284, −6.81037967825024237610727285192, −6.10007425065852311999324584480, −5.48098123951517683375651191424, −4.35197538657814850995748007851, −4.15827534817712984578069155889, −2.42959557287047146324245186301, −1.91628360146346644892156716107,
1.91628360146346644892156716107, 2.42959557287047146324245186301, 4.15827534817712984578069155889, 4.35197538657814850995748007851, 5.48098123951517683375651191424, 6.10007425065852311999324584480, 6.81037967825024237610727285192, 6.85060009229583151514173711284, 8.353545475517416957628422948266, 8.535061726195564870590584830362, 9.166064641133274764946146751231, 9.596763812420851867230064412340, 10.37590059688490989951133950811, 10.79424484238732835664106339963, 11.30716766037086820526144366698, 11.73487884299972956846533319769, 12.75603729900143268682568947074, 12.94451275136071770986430709731, 13.40695411814678904441797055855, 13.91878415200316710239896047481
