| L(s) = 1 | − 4-s + 4·9-s − 3·11-s − 3·16-s + 7·19-s + 3·29-s + 13·31-s − 4·36-s − 12·41-s + 3·44-s − 4·49-s + 9·59-s − 8·61-s + 7·64-s + 9·71-s − 7·76-s + 25·79-s + 7·81-s − 6·89-s − 12·99-s − 3·101-s + 22·109-s − 3·116-s − 2·121-s − 13·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 4/3·9-s − 0.904·11-s − 3/4·16-s + 1.60·19-s + 0.557·29-s + 2.33·31-s − 2/3·36-s − 1.87·41-s + 0.452·44-s − 4/7·49-s + 1.17·59-s − 1.02·61-s + 7/8·64-s + 1.06·71-s − 0.802·76-s + 2.81·79-s + 7/9·81-s − 0.635·89-s − 1.20·99-s − 0.298·101-s + 2.10·109-s − 0.278·116-s − 0.181·121-s − 1.16·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364368149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364368149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.714405784059450018491106879951, −9.560650675632095801929365150603, −8.708340460778589547922733754577, −8.256662421997040899527709789946, −7.83068516119876236344578835314, −7.21421690613923632249352990680, −6.76657326440517902759119044395, −6.26848832000946526631479332826, −5.36416125984623744384178469370, −4.81772661880681712009124894417, −4.61943397233531690916742056208, −3.69943729312661629930715778034, −3.03927109359466738780355277035, −2.13329041065289037363319597800, −0.977796594477563565654449984027,
0.977796594477563565654449984027, 2.13329041065289037363319597800, 3.03927109359466738780355277035, 3.69943729312661629930715778034, 4.61943397233531690916742056208, 4.81772661880681712009124894417, 5.36416125984623744384178469370, 6.26848832000946526631479332826, 6.76657326440517902759119044395, 7.21421690613923632249352990680, 7.83068516119876236344578835314, 8.256662421997040899527709789946, 8.708340460778589547922733754577, 9.560650675632095801929365150603, 9.714405784059450018491106879951
