| L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s − 6·13-s − 4·17-s + 8·19-s + 6·23-s + 3·25-s − 2·29-s − 8·31-s − 4·35-s − 8·37-s − 10·41-s + 10·43-s + 8·47-s − 8·49-s − 16·53-s − 4·55-s + 6·59-s + 10·61-s − 12·65-s − 12·67-s − 6·71-s − 10·73-s + 4·77-s + 4·79-s − 2·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s − 1.66·13-s − 0.970·17-s + 1.83·19-s + 1.25·23-s + 3/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s − 1.31·37-s − 1.56·41-s + 1.52·43-s + 1.16·47-s − 8/7·49-s − 2.19·53-s − 0.539·55-s + 0.781·59-s + 1.28·61-s − 1.48·65-s − 1.46·67-s − 0.712·71-s − 1.17·73-s + 0.455·77-s + 0.450·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18662400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18662400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 51 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 104 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 143 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 144 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 232 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 186 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.996976697974280379352762870080, −7.921241425092311591720054153332, −7.30965803989172671730917226506, −7.01466699710500967991293185219, −6.81605679435334625598929201639, −6.62588310763919330132991433914, −5.69697990901681653432427581999, −5.60293322024917231365789628723, −5.36572754146350749244203889348, −4.91840841041224054211361754056, −4.59639406326329515959608016525, −3.98452354267102359517795246236, −3.48885521211283544508025469288, −2.98709244994079036482967147787, −2.66049510613411757142832231183, −2.44077145671352364575613625238, −1.45413180828099421568742049753, −1.45193648404109700889204593146, 0, 0,
1.45193648404109700889204593146, 1.45413180828099421568742049753, 2.44077145671352364575613625238, 2.66049510613411757142832231183, 2.98709244994079036482967147787, 3.48885521211283544508025469288, 3.98452354267102359517795246236, 4.59639406326329515959608016525, 4.91840841041224054211361754056, 5.36572754146350749244203889348, 5.60293322024917231365789628723, 5.69697990901681653432427581999, 6.62588310763919330132991433914, 6.81605679435334625598929201639, 7.01466699710500967991293185219, 7.30965803989172671730917226506, 7.921241425092311591720054153332, 7.996976697974280379352762870080
