| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s + 9-s − 2·11-s − 8·14-s + 5·16-s + 2·18-s − 4·22-s + 8·23-s − 6·25-s − 12·28-s + 12·29-s + 6·32-s + 3·36-s + 12·37-s + 8·43-s − 6·44-s + 16·46-s + 9·49-s − 12·50-s + 4·53-s − 16·56-s + 24·58-s − 4·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s + 1/3·9-s − 0.603·11-s − 2.13·14-s + 5/4·16-s + 0.471·18-s − 0.852·22-s + 1.66·23-s − 6/5·25-s − 2.26·28-s + 2.22·29-s + 1.06·32-s + 1/2·36-s + 1.97·37-s + 1.21·43-s − 0.904·44-s + 2.35·46-s + 9/7·49-s − 1.69·50-s + 0.549·53-s − 2.13·56-s + 3.15·58-s − 0.503·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.656089994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.656089994\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.133011965944853566663369401893, −8.536576944084822666410557358297, −7.942118963675615622086839810216, −7.24765397306950965920878295914, −7.16399249614613621688519068074, −6.37494925689589999672182274148, −6.15685276163708486295790955262, −5.65916085944459497777328619337, −5.03220048416044134767486652990, −4.39046728862797743051098847217, −4.10666261147659159662524014932, −3.18219271670544077719825259419, −2.90847219285821389032814298970, −2.34929413803355936173509399096, −0.994198775851825156746386832320,
0.994198775851825156746386832320, 2.34929413803355936173509399096, 2.90847219285821389032814298970, 3.18219271670544077719825259419, 4.10666261147659159662524014932, 4.39046728862797743051098847217, 5.03220048416044134767486652990, 5.65916085944459497777328619337, 6.15685276163708486295790955262, 6.37494925689589999672182274148, 7.16399249614613621688519068074, 7.24765397306950965920878295914, 7.942118963675615622086839810216, 8.536576944084822666410557358297, 9.133011965944853566663369401893
