| L(s) = 1 | − 2·3-s − 4-s + 4·5-s + 9-s − 11-s + 2·12-s − 8·15-s − 4·20-s − 2·23-s + 10·25-s − 2·31-s + 2·33-s − 36-s − 2·37-s + 44-s + 4·45-s − 2·47-s + 4·49-s − 2·53-s − 4·55-s + 3·59-s + 8·60-s + 3·67-s + 4·69-s − 2·71-s − 20·75-s − 2·89-s + ⋯ |
| L(s) = 1 | − 2·3-s − 4-s + 4·5-s + 9-s − 11-s + 2·12-s − 8·15-s − 4·20-s − 2·23-s + 10·25-s − 2·31-s + 2·33-s − 36-s − 2·37-s + 44-s + 4·45-s − 2·47-s + 4·49-s − 2·53-s − 4·55-s + 3·59-s + 8·60-s + 3·67-s + 4·69-s − 2·71-s − 20·75-s − 2·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2363887025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2363887025\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 3 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 37 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−8.940220468962526153965573115303, −8.789410895450143605788855432342, −8.593747755220393752009448148288, −8.333079195952550185148115903391, −7.991086867789622509078585046695, −7.44732856757525946572974314846, −7.09269448947558754802974034129, −6.99382123070384537472379641968, −6.66195618170165305168313605544, −6.29178730509795122315024471355, −6.11508965505815348293830258521, −5.89767346726350481121266210590, −5.62726119111191211064809908129, −5.45330071038840738205166126926, −5.20760881093409547836865996317, −5.15511674058048184186949946256, −5.04570630052482461128376052956, −4.20808554678611797113438563828, −4.15808563840224811140046386721, −3.49908692231684940212974283131, −3.02902023285826984053992472111, −2.37703672066448835708596670930, −2.29900964482964363470280080739, −1.85286371698059584950090049186, −1.36827022788482007749117835448,
1.36827022788482007749117835448, 1.85286371698059584950090049186, 2.29900964482964363470280080739, 2.37703672066448835708596670930, 3.02902023285826984053992472111, 3.49908692231684940212974283131, 4.15808563840224811140046386721, 4.20808554678611797113438563828, 5.04570630052482461128376052956, 5.15511674058048184186949946256, 5.20760881093409547836865996317, 5.45330071038840738205166126926, 5.62726119111191211064809908129, 5.89767346726350481121266210590, 6.11508965505815348293830258521, 6.29178730509795122315024471355, 6.66195618170165305168313605544, 6.99382123070384537472379641968, 7.09269448947558754802974034129, 7.44732856757525946572974314846, 7.991086867789622509078585046695, 8.333079195952550185148115903391, 8.593747755220393752009448148288, 8.789410895450143605788855432342, 8.940220468962526153965573115303
