| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s − 2·5-s + 4·6-s + 9-s − 4·10-s + 4·12-s − 4·15-s − 4·16-s + 2·18-s + 4·19-s − 4·20-s + 4·23-s − 25-s − 4·27-s − 12·29-s − 8·30-s − 8·32-s + 2·36-s + 8·38-s − 4·43-s − 2·45-s + 8·46-s + 4·47-s − 8·48-s + 2·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s + 1/3·9-s − 1.26·10-s + 1.15·12-s − 1.03·15-s − 16-s + 0.471·18-s + 0.917·19-s − 0.894·20-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 2.22·29-s − 1.46·30-s − 1.41·32-s + 1/3·36-s + 1.29·38-s − 0.609·43-s − 0.298·45-s + 1.17·46-s + 0.583·47-s − 1.15·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.447944375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.447944375\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.32953415361975533969010592847, −11.00432680012034744428057104632, −10.01032633154648714313227924821, −9.377525227128497115591167438623, −8.946198865125305794237908600982, −8.366498780944735279626615226493, −7.57939344550043849739398939018, −7.33340154768720822109727888064, −6.54164836202177970269621527130, −5.52225303388555092624124767040, −5.25979913801770254313933608854, −4.09285741705128779810661003755, −3.76922697350795813379161775899, −3.11978737640806929868192726095, −2.23236505279691350596522634765,
2.23236505279691350596522634765, 3.11978737640806929868192726095, 3.76922697350795813379161775899, 4.09285741705128779810661003755, 5.25979913801770254313933608854, 5.52225303388555092624124767040, 6.54164836202177970269621527130, 7.33340154768720822109727888064, 7.57939344550043849739398939018, 8.366498780944735279626615226493, 8.946198865125305794237908600982, 9.377525227128497115591167438623, 10.01032633154648714313227924821, 11.00432680012034744428057104632, 11.32953415361975533969010592847
