| L(s) = 1 | + 3·4-s + 6·7-s − 4·13-s + 5·16-s + 2·19-s + 18·28-s − 6·31-s − 2·37-s + 2·43-s + 13·49-s − 12·52-s + 14·61-s + 3·64-s + 24·67-s − 22·73-s + 6·76-s + 30·79-s − 24·91-s − 14·97-s − 14·103-s + 34·109-s + 30·112-s + 6·121-s − 18·124-s + 127-s + 131-s + 12·133-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2.26·7-s − 1.10·13-s + 5/4·16-s + 0.458·19-s + 3.40·28-s − 1.07·31-s − 0.328·37-s + 0.304·43-s + 13/7·49-s − 1.66·52-s + 1.79·61-s + 3/8·64-s + 2.93·67-s − 2.57·73-s + 0.688·76-s + 3.37·79-s − 2.51·91-s − 1.42·97-s − 1.37·103-s + 3.25·109-s + 2.83·112-s + 6/11·121-s − 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 1.04·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.703872559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.703872559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−11.00689569522544376144711965536, −10.29782179163196724125717137600, −10.06671486536144580112316693005, −9.476891964606973669092818124945, −8.899608505640944065032256426165, −8.417124304482773719605893643125, −7.955187968227007460564027584057, −7.58795597703272934059249666996, −7.38888700808618748687935596601, −6.76241562083693148674604529358, −6.46416894052052834655430003192, −5.57443270173179946208991967374, −5.28988083268399132293722298205, −4.95326490450250652358201582254, −4.28374934155972310427296900252, −3.64271201299328746371537786528, −2.86262809521449375676695974434, −2.04797652867607365880276607097, −2.03451897170989253356148732431, −1.10101715408560923160360372720,
1.10101715408560923160360372720, 2.03451897170989253356148732431, 2.04797652867607365880276607097, 2.86262809521449375676695974434, 3.64271201299328746371537786528, 4.28374934155972310427296900252, 4.95326490450250652358201582254, 5.28988083268399132293722298205, 5.57443270173179946208991967374, 6.46416894052052834655430003192, 6.76241562083693148674604529358, 7.38888700808618748687935596601, 7.58795597703272934059249666996, 7.955187968227007460564027584057, 8.417124304482773719605893643125, 8.899608505640944065032256426165, 9.476891964606973669092818124945, 10.06671486536144580112316693005, 10.29782179163196724125717137600, 11.00689569522544376144711965536
