| L(s) = 1 | + 2·7-s + 2·11-s − 2·13-s − 3·17-s − 3·19-s + 3·23-s − 8·29-s − 5·31-s + 8·37-s + 6·41-s − 20·43-s + 9·47-s − 6·49-s − 53-s − 6·59-s − 11·61-s + 6·67-s − 14·71-s − 2·73-s + 4·77-s − 11·79-s + 83-s + 20·89-s − 4·91-s − 6·97-s + 6·101-s + 12·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.603·11-s − 0.554·13-s − 0.727·17-s − 0.688·19-s + 0.625·23-s − 1.48·29-s − 0.898·31-s + 1.31·37-s + 0.937·41-s − 3.04·43-s + 1.31·47-s − 6/7·49-s − 0.137·53-s − 0.781·59-s − 1.40·61-s + 0.733·67-s − 1.66·71-s − 0.234·73-s + 0.455·77-s − 1.23·79-s + 0.109·83-s + 2.11·89-s − 0.419·91-s − 0.609·97-s + 0.597·101-s + 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 35 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 113 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 95 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 187 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 105 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.54950892259714879953539379463, −7.35972990049802873612318575040, −6.84092419297923384874577721699, −6.55613449499843370710943959631, −6.22657661393673910892764474823, −5.94423965647449196866592912819, −5.37805617151707899110941103486, −5.17109985595309513323391077084, −4.71911471802450880915649858793, −4.50633739897342558499392459835, −4.00623068770582199970164609822, −3.80997504540090385577158650842, −3.18774796290811662273099437258, −2.91042644258290272161692912629, −2.17432288050964842618024054494, −2.13091268542081583844267243785, −1.34817922856538958334166454247, −1.27791120627764478213900858438, 0, 0,
1.27791120627764478213900858438, 1.34817922856538958334166454247, 2.13091268542081583844267243785, 2.17432288050964842618024054494, 2.91042644258290272161692912629, 3.18774796290811662273099437258, 3.80997504540090385577158650842, 4.00623068770582199970164609822, 4.50633739897342558499392459835, 4.71911471802450880915649858793, 5.17109985595309513323391077084, 5.37805617151707899110941103486, 5.94423965647449196866592912819, 6.22657661393673910892764474823, 6.55613449499843370710943959631, 6.84092419297923384874577721699, 7.35972990049802873612318575040, 7.54950892259714879953539379463
