| L(s) = 1 | + 3-s + 5-s − 5·7-s + 3·9-s + 6·11-s + 4·13-s + 15-s + 6·17-s + 8·19-s − 5·21-s + 3·23-s + 8·27-s + 6·29-s + 2·31-s + 6·33-s − 5·35-s − 8·37-s + 4·39-s − 6·41-s − 10·43-s + 3·45-s + 18·49-s + 6·51-s − 12·53-s + 6·55-s + 8·57-s + 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.88·7-s + 9-s + 1.80·11-s + 1.10·13-s + 0.258·15-s + 1.45·17-s + 1.83·19-s − 1.09·21-s + 0.625·23-s + 1.53·27-s + 1.11·29-s + 0.359·31-s + 1.04·33-s − 0.845·35-s − 1.31·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s + 0.447·45-s + 18/7·49-s + 0.840·51-s − 1.64·53-s + 0.809·55-s + 1.05·57-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.888152887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.888152887\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.68989361494849463274435359719, −10.46058670377058933316801826783, −9.786938663600683787192105588512, −9.786880525507366575165268719843, −9.301038899712288362920988552275, −8.998119376833668451673315465199, −8.363417278521026213757216681673, −8.021720856044280653995378443389, −7.12329146953734365226760007333, −6.88162324167860436496306637714, −6.55582359669875240872898298476, −6.17272885613011890655167590188, −5.45673934731770641113599573184, −4.99720724392104398638989585932, −4.07725842477931560750023000586, −3.63128004650993655204502494964, −3.15177141816932700949011503777, −2.94219701204365767507581915833, −1.32923166829179909689328365356, −1.27941598824344767311977059718,
1.27941598824344767311977059718, 1.32923166829179909689328365356, 2.94219701204365767507581915833, 3.15177141816932700949011503777, 3.63128004650993655204502494964, 4.07725842477931560750023000586, 4.99720724392104398638989585932, 5.45673934731770641113599573184, 6.17272885613011890655167590188, 6.55582359669875240872898298476, 6.88162324167860436496306637714, 7.12329146953734365226760007333, 8.021720856044280653995378443389, 8.363417278521026213757216681673, 8.998119376833668451673315465199, 9.301038899712288362920988552275, 9.786880525507366575165268719843, 9.786938663600683787192105588512, 10.46058670377058933316801826783, 10.68989361494849463274435359719
