| L(s) = 1 | − 2-s − 5-s + 2·7-s + 8-s − 9-s + 10-s + 11-s − 2·13-s − 2·14-s − 16-s + 18-s + 19-s − 22-s + 23-s + 2·26-s − 2·35-s + 37-s − 38-s − 40-s − 2·41-s + 45-s − 46-s + 47-s + 3·49-s + 53-s − 55-s + 2·56-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 2·7-s + 8-s − 9-s + 10-s + 11-s − 2·13-s − 2·14-s − 16-s + 18-s + 19-s − 22-s + 23-s + 2·26-s − 2·35-s + 37-s − 38-s − 40-s − 2·41-s + 45-s − 46-s + 47-s + 3·49-s + 53-s − 55-s + 2·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3243421534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3243421534\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.13830326851865339363536546666, −11.80071508954180853926845378276, −11.20615371565957059761013023842, −11.13932298076329041515698727402, −10.48062927226413804045761639682, −9.860082458716909779214524538077, −9.368086750882871791009580674703, −8.973991931372176462136622755714, −8.292953158098930586523783564286, −8.244697824609083528068906561575, −7.45877146856009408561287802246, −7.38314043813070187558221167778, −6.79736537383575133286069677998, −5.53551928462064910010219100853, −5.26808382279490391258148059126, −4.45199213217626417055744200907, −4.38038338526091381188467776575, −3.23446085663219780964219393327, −2.30247443887687932688881450063, −1.29490937665423030484310327694,
1.29490937665423030484310327694, 2.30247443887687932688881450063, 3.23446085663219780964219393327, 4.38038338526091381188467776575, 4.45199213217626417055744200907, 5.26808382279490391258148059126, 5.53551928462064910010219100853, 6.79736537383575133286069677998, 7.38314043813070187558221167778, 7.45877146856009408561287802246, 8.244697824609083528068906561575, 8.292953158098930586523783564286, 8.973991931372176462136622755714, 9.368086750882871791009580674703, 9.860082458716909779214524538077, 10.48062927226413804045761639682, 11.13932298076329041515698727402, 11.20615371565957059761013023842, 11.80071508954180853926845378276, 12.13830326851865339363536546666
