| L(s) = 1 | + 2·3-s − 4·5-s + 9-s − 4·11-s + 16·13-s − 8·15-s − 4·17-s − 4·23-s + 12·25-s − 2·27-s − 16·29-s − 8·31-s − 8·33-s + 8·37-s + 32·39-s + 8·41-s − 4·45-s − 8·51-s + 4·53-s + 16·55-s − 8·59-s − 16·61-s − 64·65-s − 8·69-s + 8·71-s − 8·73-s + 24·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s + 4.43·13-s − 2.06·15-s − 0.970·17-s − 0.834·23-s + 12/5·25-s − 0.384·27-s − 2.97·29-s − 1.43·31-s − 1.39·33-s + 1.31·37-s + 5.12·39-s + 1.24·41-s − 0.596·45-s − 1.12·51-s + 0.549·53-s + 2.15·55-s − 1.04·59-s − 2.04·61-s − 7.93·65-s − 0.963·69-s + 0.949·71-s − 0.936·73-s + 2.77·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655263607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655263607\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 56 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 112 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} + 64 T^{3} + 1955 T^{4} + 64 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 64 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 736 p T^{5} + 88 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 102 T^{2} + 5915 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T - 656 T^{3} - 5905 T^{4} - 656 p T^{5} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 66 T^{2} - 512 T^{3} + 9635 T^{4} - 512 p T^{5} + 66 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 1640 p T^{5} + 140 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.25501695047126728192532491829, −6.12828046890969184957813607322, −6.08832715504166373699955198292, −5.77608615893460746556427438818, −5.74059126780307498440962963580, −5.26464166278851014294869468191, −5.09463882897299474946038088414, −4.95532838166331572820802489256, −4.49222365447608573408195898954, −4.33370124531380089695385918679, −4.03890435393439115046861694035, −3.99076366076338142621916970024, −3.76603456855653435810253070265, −3.57813896172086589162832373592, −3.33196213394815158410855379853, −3.21405128954750419942968365275, −3.13446574965654026121654987078, −2.51973511172944481287662037351, −2.32876643329338936231765442298, −2.05118661902886876074088602030, −1.84515130578861641669598011811, −1.36358588209344837365117866681, −1.16205599674845513647561804054, −0.70887067804965843296346965018, −0.22519150083049940317497307569,
0.22519150083049940317497307569, 0.70887067804965843296346965018, 1.16205599674845513647561804054, 1.36358588209344837365117866681, 1.84515130578861641669598011811, 2.05118661902886876074088602030, 2.32876643329338936231765442298, 2.51973511172944481287662037351, 3.13446574965654026121654987078, 3.21405128954750419942968365275, 3.33196213394815158410855379853, 3.57813896172086589162832373592, 3.76603456855653435810253070265, 3.99076366076338142621916970024, 4.03890435393439115046861694035, 4.33370124531380089695385918679, 4.49222365447608573408195898954, 4.95532838166331572820802489256, 5.09463882897299474946038088414, 5.26464166278851014294869468191, 5.74059126780307498440962963580, 5.77608615893460746556427438818, 6.08832715504166373699955198292, 6.12828046890969184957813607322, 6.25501695047126728192532491829
