| L(s) = 1 | + 4·5-s + 4·13-s + 12·17-s + 2·25-s + 4·29-s + 4·37-s − 4·41-s + 2·49-s + 20·53-s − 12·61-s + 16·65-s − 12·73-s + 48·85-s − 20·89-s − 28·97-s − 12·101-s − 28·109-s − 4·113-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.10·13-s + 2.91·17-s + 2/5·25-s + 0.742·29-s + 0.657·37-s − 0.624·41-s + 2/7·49-s + 2.74·53-s − 1.53·61-s + 1.98·65-s − 1.40·73-s + 5.20·85-s − 2.11·89-s − 2.84·97-s − 1.19·101-s − 2.68·109-s − 0.376·113-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.127152398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.127152398\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.844581178668094717437191348011, −8.131897207027853144179789700526, −8.093022628739907221893201157530, −7.33135946825865482348621844071, −6.83791452023955841983204630481, −6.23593377615022317750995350027, −5.82131382158921014822362291005, −5.42460705368026111370695370006, −5.37335304530331906156568812203, −4.20875443528614711634884665822, −3.81654902962345598726342363363, −2.97479066673588337231673992698, −2.59447091927996366543258857368, −1.47354540780337201919328149138, −1.26883614185569776706713350485,
1.26883614185569776706713350485, 1.47354540780337201919328149138, 2.59447091927996366543258857368, 2.97479066673588337231673992698, 3.81654902962345598726342363363, 4.20875443528614711634884665822, 5.37335304530331906156568812203, 5.42460705368026111370695370006, 5.82131382158921014822362291005, 6.23593377615022317750995350027, 6.83791452023955841983204630481, 7.33135946825865482348621844071, 8.093022628739907221893201157530, 8.131897207027853144179789700526, 8.844581178668094717437191348011
