| L(s) = 1 | − 2-s − 4-s + 3·8-s + 8·11-s − 16-s − 8·22-s − 10·25-s − 5·32-s + 24·43-s − 8·44-s − 7·49-s + 10·50-s + 7·64-s + 8·67-s − 24·86-s + 24·88-s + 7·98-s + 10·100-s + 40·107-s + 4·113-s + 26·121-s + 127-s + 3·128-s + 131-s − 8·134-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2.41·11-s − 1/4·16-s − 1.70·22-s − 2·25-s − 0.883·32-s + 3.65·43-s − 1.20·44-s − 49-s + 1.41·50-s + 7/8·64-s + 0.977·67-s − 2.58·86-s + 2.55·88-s + 0.707·98-s + 100-s + 3.86·107-s + 0.376·113-s + 2.36·121-s + 0.0887·127-s + 0.265·128-s + 0.0873·131-s − 0.691·134-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165430910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165430910\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−11.20961909287264512335255494150, −10.68269158535820971643676472078, −9.967431838943688016049105126496, −9.791990885502652367938957204653, −9.214816801595176800766717039881, −9.121566261915184788051845778858, −8.625903801365671790590457509253, −8.036103551643299844878937876337, −7.57992102978520621740334554130, −7.21622080974161626884662623880, −6.59623147254894945359251602149, −5.99081122971328745865502444770, −5.80431869535159251952096146703, −4.85560911127170109792540992087, −4.31315715256686274165587589135, −3.89518728377610978818893024479, −3.53264502831397871108854111783, −2.31471174308058104799078855800, −1.59107872405238015477568672891, −0.806097709578990692063738973092,
0.806097709578990692063738973092, 1.59107872405238015477568672891, 2.31471174308058104799078855800, 3.53264502831397871108854111783, 3.89518728377610978818893024479, 4.31315715256686274165587589135, 4.85560911127170109792540992087, 5.80431869535159251952096146703, 5.99081122971328745865502444770, 6.59623147254894945359251602149, 7.21622080974161626884662623880, 7.57992102978520621740334554130, 8.036103551643299844878937876337, 8.625903801365671790590457509253, 9.121566261915184788051845778858, 9.214816801595176800766717039881, 9.791990885502652367938957204653, 9.967431838943688016049105126496, 10.68269158535820971643676472078, 11.20961909287264512335255494150
