| L(s) = 1 | − 5-s − 2·7-s − 4·11-s + 2·13-s + 10·17-s − 10·19-s − 23-s + 2·29-s − 7·31-s + 2·35-s − 12·37-s − 4·43-s − 4·47-s + 7·49-s + 18·53-s + 4·55-s − 14·59-s + 11·61-s − 2·65-s − 14·67-s − 24·73-s + 8·77-s + 3·79-s + 83-s − 10·85-s − 4·91-s + 10·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.554·13-s + 2.42·17-s − 2.29·19-s − 0.208·23-s + 0.371·29-s − 1.25·31-s + 0.338·35-s − 1.97·37-s − 0.609·43-s − 0.583·47-s + 49-s + 2.47·53-s + 0.539·55-s − 1.82·59-s + 1.40·61-s − 0.248·65-s − 1.71·67-s − 2.80·73-s + 0.911·77-s + 0.337·79-s + 0.109·83-s − 1.08·85-s − 0.419·91-s + 1.02·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 16 T + 159 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.426214136173549626823674681205, −8.221461241786966854261183429566, −7.74102598348565152369449584586, −7.34717601267661802066936726957, −6.93475415823980426743478386688, −6.79260929886829257536748545985, −6.04424274637735167775994557677, −5.84089177059906593529320536819, −5.40564780647186887300642396942, −5.24571667797557006126350836153, −4.47582306442265481942921054047, −4.13132947735132223594679723017, −3.61993608192267871246527396146, −3.43473305800243617601064676474, −2.75138002872948096609710356761, −2.52708640949624681869371476409, −1.66291354796525308838072874263, −1.26675937112357791425143162767, 0, 0,
1.26675937112357791425143162767, 1.66291354796525308838072874263, 2.52708640949624681869371476409, 2.75138002872948096609710356761, 3.43473305800243617601064676474, 3.61993608192267871246527396146, 4.13132947735132223594679723017, 4.47582306442265481942921054047, 5.24571667797557006126350836153, 5.40564780647186887300642396942, 5.84089177059906593529320536819, 6.04424274637735167775994557677, 6.79260929886829257536748545985, 6.93475415823980426743478386688, 7.34717601267661802066936726957, 7.74102598348565152369449584586, 8.221461241786966854261183429566, 8.426214136173549626823674681205
