| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 8·5-s − 4·6-s − 2·7-s − 2·8-s + 9-s − 16·10-s + 6·11-s − 2·12-s − 4·14-s + 16·15-s − 4·16-s + 8·17-s + 2·18-s + 6·19-s − 8·20-s + 4·21-s + 12·22-s + 2·23-s + 4·24-s + 26·25-s + 2·27-s − 2·28-s + 2·29-s + 32·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 3.57·5-s − 1.63·6-s − 0.755·7-s − 0.707·8-s + 1/3·9-s − 5.05·10-s + 1.80·11-s − 0.577·12-s − 1.06·14-s + 4.13·15-s − 16-s + 1.94·17-s + 0.471·18-s + 1.37·19-s − 1.78·20-s + 0.872·21-s + 2.55·22-s + 0.417·23-s + 0.816·24-s + 26/5·25-s + 0.384·27-s − 0.377·28-s + 0.371·29-s + 5.84·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570441674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570441674\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 T - 8 T^{2} - 4 T^{3} + 67 T^{4} - 4 p T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 14 T + 85 T^{2} - 14 p T^{3} + 100 p T^{4} - 14 p^{2} T^{5} + 85 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2 T + 29 T^{2} + 214 T^{3} - 1220 T^{4} + 214 p T^{5} + 29 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 148 T^{3} - 1877 T^{4} + 148 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T + 16 T^{2} + 292 T^{3} - 4613 T^{4} + 292 p T^{5} + 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T - 112 T^{2} - 36 T^{3} + 14307 T^{4} - 36 p T^{5} - 112 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 16 T + 207 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 58 T^{2} + 288 T^{3} + 20067 T^{4} + 288 p T^{5} - 58 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.14128121001702942475010814322, −6.78144416351220564778474420179, −6.76226750627744034975794504657, −6.58670475577782700437598061361, −5.98963947299702177873416977959, −5.97133185816806069970240002975, −5.60973154833169905546254170016, −5.54892460857752470125586669361, −5.48743130903330395770228480388, −4.89472731840353758347188571777, −4.76707202453011788284952375861, −4.34265723533652992425327322135, −4.23871446153835812201102453637, −4.19876384096816739662899619372, −3.80803947700986996825766035510, −3.64595289559704178267974483901, −3.62237777737511433837477943384, −3.24286769883718647623318000336, −2.84527503616467055042524689961, −2.69555424734731355073076774919, −2.43066041189292479433514768450, −1.32568421983442431747852034371, −1.06067765641103762872017077809, −0.801631083450501054090742212136, −0.40465708441461393214548132699,
0.40465708441461393214548132699, 0.801631083450501054090742212136, 1.06067765641103762872017077809, 1.32568421983442431747852034371, 2.43066041189292479433514768450, 2.69555424734731355073076774919, 2.84527503616467055042524689961, 3.24286769883718647623318000336, 3.62237777737511433837477943384, 3.64595289559704178267974483901, 3.80803947700986996825766035510, 4.19876384096816739662899619372, 4.23871446153835812201102453637, 4.34265723533652992425327322135, 4.76707202453011788284952375861, 4.89472731840353758347188571777, 5.48743130903330395770228480388, 5.54892460857752470125586669361, 5.60973154833169905546254170016, 5.97133185816806069970240002975, 5.98963947299702177873416977959, 6.58670475577782700437598061361, 6.76226750627744034975794504657, 6.78144416351220564778474420179, 7.14128121001702942475010814322
