| L(s) = 1 | − 2-s + 3-s − 4·5-s − 6-s + 4·10-s − 4·15-s + 5·19-s + 3·23-s + 10·25-s + 4·30-s − 31-s + 32-s − 5·38-s − 3·46-s + 49-s − 10·50-s + 5·57-s + 62-s − 64-s + 3·69-s + 10·75-s + 2·79-s − 2·83-s − 93-s − 20·95-s + 96-s − 98-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 4·5-s − 6-s + 4·10-s − 4·15-s + 5·19-s + 3·23-s + 10·25-s + 4·30-s − 31-s + 32-s − 5·38-s − 3·46-s + 49-s − 10·50-s + 5·57-s + 62-s − 64-s + 3·69-s + 10·75-s + 2·79-s − 2·83-s − 93-s − 20·95-s + 96-s − 98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5111042262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5111042262\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 7 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 83 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 89 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−7.05233650206445582930015938585, −6.80625363589277479611478844496, −6.57621769153610561090583422930, −6.46186670530347118588049342567, −5.81127512649439998327069896776, −5.48750284093002915260974169871, −5.42868939354976063403352338767, −5.28894540769403650443196585783, −4.97655005481667650370630086224, −4.83404655929998743129222863774, −4.48428233879645075623191701894, −4.35711757772140610690028697234, −4.19117222202995620490057919495, −3.70491334451218341265604651568, −3.55078957551036625109132315300, −3.30891223862349533646683871086, −3.24364145005132855106014252961, −3.06760707477638920498439212994, −2.81769227945297205556446243812, −2.79535548866803986971743982040, −2.18009902994333449570581966654, −1.49575708439865911393809218343, −1.03902734516361877238058269078, −0.852103272735277286414273935796, −0.76428550921295347572565042270,
0.76428550921295347572565042270, 0.852103272735277286414273935796, 1.03902734516361877238058269078, 1.49575708439865911393809218343, 2.18009902994333449570581966654, 2.79535548866803986971743982040, 2.81769227945297205556446243812, 3.06760707477638920498439212994, 3.24364145005132855106014252961, 3.30891223862349533646683871086, 3.55078957551036625109132315300, 3.70491334451218341265604651568, 4.19117222202995620490057919495, 4.35711757772140610690028697234, 4.48428233879645075623191701894, 4.83404655929998743129222863774, 4.97655005481667650370630086224, 5.28894540769403650443196585783, 5.42868939354976063403352338767, 5.48750284093002915260974169871, 5.81127512649439998327069896776, 6.46186670530347118588049342567, 6.57621769153610561090583422930, 6.80625363589277479611478844496, 7.05233650206445582930015938585
