L(s) = 1 | − 9-s + 8·11-s + 8·19-s − 4·29-s + 8·31-s + 4·41-s − 2·49-s + 8·59-s + 12·61-s − 32·71-s − 8·79-s + 81-s − 20·89-s − 8·99-s − 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 2.41·11-s + 1.83·19-s − 0.742·29-s + 1.43·31-s + 0.624·41-s − 2/7·49-s + 1.04·59-s + 1.53·61-s − 3.79·71-s − 0.900·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s − 1.19·101-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.206898949\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.206898949\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.172334598196665533405479206429, −8.730371025367728914499779785731, −8.587951252016273637338795760782, −7.929082657623140474806994641005, −7.70483590849826398452023744391, −7.07445107945197400748860515084, −6.75238372979504293534515700626, −6.71636773570162388410779518654, −5.91424571996545877171622419745, −5.65721675660074823230685430187, −5.45293474935122134271863367147, −4.58080055864319745795182284075, −4.37918311759566941644964555042, −3.89303030399518719000952706235, −3.51242444653352122538651643465, −2.90911767656121215086916233827, −2.61720373801460377413289819432, −1.51872077083155707394457387974, −1.42547358060366386854094773074, −0.66339537121305672199305319677,
0.66339537121305672199305319677, 1.42547358060366386854094773074, 1.51872077083155707394457387974, 2.61720373801460377413289819432, 2.90911767656121215086916233827, 3.51242444653352122538651643465, 3.89303030399518719000952706235, 4.37918311759566941644964555042, 4.58080055864319745795182284075, 5.45293474935122134271863367147, 5.65721675660074823230685430187, 5.91424571996545877171622419745, 6.71636773570162388410779518654, 6.75238372979504293534515700626, 7.07445107945197400748860515084, 7.70483590849826398452023744391, 7.929082657623140474806994641005, 8.587951252016273637338795760782, 8.730371025367728914499779785731, 9.172334598196665533405479206429