| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s − 4·8-s + 3·9-s + 2·10-s − 3·11-s + 3·12-s − 5·13-s − 15-s + 5·16-s + 2·17-s − 6·18-s − 5·19-s − 3·20-s + 6·22-s − 6·23-s − 4·24-s − 25-s + 10·26-s + 8·27-s + 3·29-s + 2·30-s − 6·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s − 1.41·8-s + 9-s + 0.632·10-s − 0.904·11-s + 0.866·12-s − 1.38·13-s − 0.258·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 1.14·19-s − 0.670·20-s + 1.27·22-s − 1.25·23-s − 0.816·24-s − 1/5·25-s + 1.96·26-s + 1.53·27-s + 0.557·29-s + 0.365·30-s − 1.07·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 90 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 116 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 144 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 206 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 164 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.151088353727058829378633377593, −7.913588687712218117356791582847, −7.70416058859737732385644462708, −7.33315105433078133673816585691, −7.03164976737505907274552577369, −6.60816967283653619660568671913, −6.31305303076110485966168378273, −5.66570281524977280925037679614, −5.40440521408238427809354762405, −4.74402882647823446725398486014, −4.50800272966772009184168152681, −4.00947163142639427802969339758, −3.43338595660807186451950297117, −3.08189354489351561666007719885, −2.39318951590849486040966270499, −2.29203758292391617207312838769, −1.67955102688756535654849862759, −1.13274480395633697769138563949, 0, 0,
1.13274480395633697769138563949, 1.67955102688756535654849862759, 2.29203758292391617207312838769, 2.39318951590849486040966270499, 3.08189354489351561666007719885, 3.43338595660807186451950297117, 4.00947163142639427802969339758, 4.50800272966772009184168152681, 4.74402882647823446725398486014, 5.40440521408238427809354762405, 5.66570281524977280925037679614, 6.31305303076110485966168378273, 6.60816967283653619660568671913, 7.03164976737505907274552577369, 7.33315105433078133673816585691, 7.70416058859737732385644462708, 7.913588687712218117356791582847, 8.151088353727058829378633377593
