| L(s) = 1 | + 2·2-s + 3-s + 4-s − 4·5-s + 2·6-s − 8·10-s + 12-s − 4·15-s + 2·17-s + 3·19-s − 4·20-s − 3·23-s + 10·25-s − 8·30-s − 31-s − 2·32-s + 4·34-s + 6·38-s − 6·46-s + 2·47-s − 49-s + 20·50-s + 2·51-s + 2·53-s + 3·57-s − 4·60-s − 2·61-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3-s + 4-s − 4·5-s + 2·6-s − 8·10-s + 12-s − 4·15-s + 2·17-s + 3·19-s − 4·20-s − 3·23-s + 10·25-s − 8·30-s − 31-s − 2·32-s + 4·34-s + 6·38-s − 6·46-s + 2·47-s − 49-s + 20·50-s + 2·51-s + 2·53-s + 3·57-s − 4·60-s − 2·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8423577551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8423577551\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 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 | 2 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 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_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 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_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 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
−8.200313253094735482642034836555, −7.86981536979193501296300366325, −7.66114530167905324722248674862, −7.53037687664239779805333705603, −7.32626997757935531498737623882, −7.20640943406272869220833045740, −7.05685343820241552767001254678, −6.57415191952588966007488319832, −6.00942766220173371116897722317, −5.79055349743668554015690216602, −5.60320453543794570246483467647, −5.27156868949862611646974197434, −5.02696050875600660725499687046, −4.75261768268870028687845134520, −4.49943002105415351046842674801, −4.17500236475816437904917529251, −4.15695407487105695507077282479, −3.55971439393958071159535120744, −3.54459350698007769024446441711, −3.50420858773501775128994078260, −3.34979100782801195602045840862, −2.68029501482923561067883671494, −2.56917715112723712525892423177, −1.57845791781518073905096450400, −1.00003935748774709023449833860,
1.00003935748774709023449833860, 1.57845791781518073905096450400, 2.56917715112723712525892423177, 2.68029501482923561067883671494, 3.34979100782801195602045840862, 3.50420858773501775128994078260, 3.54459350698007769024446441711, 3.55971439393958071159535120744, 4.15695407487105695507077282479, 4.17500236475816437904917529251, 4.49943002105415351046842674801, 4.75261768268870028687845134520, 5.02696050875600660725499687046, 5.27156868949862611646974197434, 5.60320453543794570246483467647, 5.79055349743668554015690216602, 6.00942766220173371116897722317, 6.57415191952588966007488319832, 7.05685343820241552767001254678, 7.20640943406272869220833045740, 7.32626997757935531498737623882, 7.53037687664239779805333705603, 7.66114530167905324722248674862, 7.86981536979193501296300366325, 8.200313253094735482642034836555
