| L(s) = 1 | + 2·2-s − 4·3-s + 4-s − 2·5-s − 8·6-s − 2·8-s + 8·9-s − 4·10-s − 4·11-s − 4·12-s − 8·13-s + 8·15-s − 4·16-s − 8·17-s + 16·18-s − 4·19-s − 2·20-s − 8·22-s + 8·23-s + 8·24-s + 25-s − 16·26-s − 8·27-s − 8·29-s + 16·30-s − 2·32-s + 16·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s − 3.26·6-s − 0.707·8-s + 8/3·9-s − 1.26·10-s − 1.20·11-s − 1.15·12-s − 2.21·13-s + 2.06·15-s − 16-s − 1.94·17-s + 3.77·18-s − 0.917·19-s − 0.447·20-s − 1.70·22-s + 1.66·23-s + 1.63·24-s + 1/5·25-s − 3.13·26-s − 1.53·27-s − 1.48·29-s + 2.92·30-s − 0.353·32-s + 2.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06461842835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06461842835\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 16 T^{2} + 112 T^{3} + 927 T^{4} + 112 p T^{5} + 16 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 24 T^{2} + 8 T^{3} + 935 T^{4} + 8 p T^{5} - 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 10 T^{2} - 64 T^{3} + 915 T^{4} - 64 p T^{5} + 10 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 54 T^{2} + 1955 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 30 T^{2} + 112 T^{3} + 155 T^{4} + 112 p T^{5} - 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 106 T^{2} + 896 T^{3} + 8259 T^{4} + 896 p T^{5} + 106 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T - 22 T^{2} + 272 T^{3} - 2213 T^{4} + 272 p T^{5} - 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 20 T + 184 T^{2} + 1960 T^{3} + 19575 T^{4} + 1960 p T^{5} + 184 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 18 T^{2} + 496 T^{3} - 4693 T^{4} + 496 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 54 T^{2} + 128 T^{3} + 5147 T^{4} + 128 p T^{5} - 54 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 48 T^{2} - 992 T^{3} + 19247 T^{4} - 992 p T^{5} + 48 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8 T - 102 T^{2} - 64 T^{3} + 16259 T^{4} - 64 p T^{5} - 102 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 152 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 16 T^{2} - 7665 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.936771867383309706891939166051, −7.57108997768208190963377909849, −7.26337250303994701588309901801, −6.82453946204241736379610225322, −6.74854506287556556884382046235, −6.72736538157589128210734524460, −6.66928124572189647309933390780, −5.99150274161858809216318371203, −5.79417591442852610881875451218, −5.58305453822436293208156145088, −5.45401877792578967598540482494, −4.94403947399960585936282373161, −4.76663202380792385127221259651, −4.74212017091794765072573619744, −4.70536813063364626810720683905, −4.34532839533486859174709888597, −3.86795775579693543455859974492, −3.56968288990767405742367342808, −3.26600088788341794648264759924, −2.79851004657707238316558551843, −2.55761865698884079924560305099, −2.08890214597006709681330387013, −1.78752174640145978597414343107, −0.68731878865112363732629202782, −0.11862930177731519879363950665,
0.11862930177731519879363950665, 0.68731878865112363732629202782, 1.78752174640145978597414343107, 2.08890214597006709681330387013, 2.55761865698884079924560305099, 2.79851004657707238316558551843, 3.26600088788341794648264759924, 3.56968288990767405742367342808, 3.86795775579693543455859974492, 4.34532839533486859174709888597, 4.70536813063364626810720683905, 4.74212017091794765072573619744, 4.76663202380792385127221259651, 4.94403947399960585936282373161, 5.45401877792578967598540482494, 5.58305453822436293208156145088, 5.79417591442852610881875451218, 5.99150274161858809216318371203, 6.66928124572189647309933390780, 6.72736538157589128210734524460, 6.74854506287556556884382046235, 6.82453946204241736379610225322, 7.26337250303994701588309901801, 7.57108997768208190963377909849, 7.936771867383309706891939166051
