| L(s) = 1 | + 7-s − 2·9-s + 16·23-s + 6·25-s + 4·29-s − 12·37-s + 16·43-s + 49-s − 20·53-s − 2·63-s − 24·67-s − 16·79-s − 5·81-s − 24·107-s + 20·109-s + 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 2/3·9-s + 3.33·23-s + 6/5·25-s + 0.742·29-s − 1.97·37-s + 2.43·43-s + 1/7·49-s − 2.74·53-s − 0.251·63-s − 2.93·67-s − 1.80·79-s − 5/9·81-s − 2.32·107-s + 1.91·109-s + 1.12·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.26·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208681114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208681114\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.85093898417625194511634818884, −10.55520673636509499483521968270, −9.590662754018899629342276160434, −8.928329697232844766585907221767, −8.870187268949863787955145092048, −8.188057214509100026281306554311, −7.22672629765934394974316502798, −7.14438434722230698367292292955, −6.27767574855681654030919817110, −5.57466416223045488855827312533, −4.90170998996837834018405216678, −4.50185552371175935345657331179, −3.14137792992390816749783057891, −2.87703403465824901690577931359, −1.30690906969982341922250798482,
1.30690906969982341922250798482, 2.87703403465824901690577931359, 3.14137792992390816749783057891, 4.50185552371175935345657331179, 4.90170998996837834018405216678, 5.57466416223045488855827312533, 6.27767574855681654030919817110, 7.14438434722230698367292292955, 7.22672629765934394974316502798, 8.188057214509100026281306554311, 8.870187268949863787955145092048, 8.928329697232844766585907221767, 9.590662754018899629342276160434, 10.55520673636509499483521968270, 10.85093898417625194511634818884
