| L(s) = 1 | + 2·5-s + 2·7-s + 4·11-s − 2·17-s − 2·19-s − 4·23-s − 2·25-s + 2·29-s − 12·31-s + 4·35-s + 8·37-s + 8·41-s + 16·43-s + 10·47-s + 3·49-s − 18·53-s + 8·55-s − 12·67-s − 2·71-s + 16·73-s + 8·77-s + 8·79-s + 14·83-s − 4·85-s − 4·95-s − 8·97-s + 26·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.20·11-s − 0.485·17-s − 0.458·19-s − 0.834·23-s − 2/5·25-s + 0.371·29-s − 2.15·31-s + 0.676·35-s + 1.31·37-s + 1.24·41-s + 2.43·43-s + 1.45·47-s + 3/7·49-s − 2.47·53-s + 1.07·55-s − 1.46·67-s − 0.237·71-s + 1.87·73-s + 0.911·77-s + 0.900·79-s + 1.53·83-s − 0.433·85-s − 0.410·95-s − 0.812·97-s + 2.58·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.142557905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.142557905\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 98 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 170 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.67522543234814574563635983307, −7.64437695385880725401287779006, −7.25019596556098432038560281367, −6.85702262341051165929962564898, −6.24525672208393597435706760113, −6.12148068225185191542316168771, −5.90132473328886452694033131563, −5.72162521374634430178265625180, −4.99843751641245220274958892778, −4.77012150595774402397662199076, −4.25135340697210115009910119665, −4.22674619226953325956924148818, −3.54231726191946630869870498148, −3.45695013805998129204336405157, −2.55154079721994403318654530298, −2.35648707723637607040628678587, −1.79815819521805936961832817336, −1.78316418100372479469732440228, −0.931721484180566830078057361923, −0.57669994224633301783839881854,
0.57669994224633301783839881854, 0.931721484180566830078057361923, 1.78316418100372479469732440228, 1.79815819521805936961832817336, 2.35648707723637607040628678587, 2.55154079721994403318654530298, 3.45695013805998129204336405157, 3.54231726191946630869870498148, 4.22674619226953325956924148818, 4.25135340697210115009910119665, 4.77012150595774402397662199076, 4.99843751641245220274958892778, 5.72162521374634430178265625180, 5.90132473328886452694033131563, 6.12148068225185191542316168771, 6.24525672208393597435706760113, 6.85702262341051165929962564898, 7.25019596556098432038560281367, 7.64437695385880725401287779006, 7.67522543234814574563635983307
