| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 8-s − 10-s − 6·11-s + 12·13-s + 15-s − 16-s + 4·19-s − 6·22-s + 24-s + 12·26-s + 27-s − 16·29-s + 30-s − 2·31-s + 6·33-s − 4·37-s + 4·38-s − 12·39-s + 40-s − 20·41-s − 12·43-s + 2·47-s + 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 0.316·10-s − 1.80·11-s + 3.32·13-s + 0.258·15-s − 1/4·16-s + 0.917·19-s − 1.27·22-s + 0.204·24-s + 2.35·26-s + 0.192·27-s − 2.97·29-s + 0.182·30-s − 0.359·31-s + 1.04·33-s − 0.657·37-s + 0.648·38-s − 1.92·39-s + 0.158·40-s − 3.12·41-s − 1.82·43-s + 0.291·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.043393454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043393454\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.850772558168846196110030715334, −9.075030076268185221531080378088, −8.955466640314035603967422821677, −8.392065745180742976954026678755, −8.206743980650072918043135650299, −7.52596873001703172416476315627, −7.49262597290169564919552252457, −6.61474828444259376711934654195, −6.40454443889734962686846198108, −5.80885774896028068653041968795, −5.63199499408319090999921245342, −5.05015336255964301849953312111, −4.98180200108526678902327097794, −4.11186049711831909206671923886, −3.55890901417255229017515382962, −3.37406758071836814924915307209, −3.11034124194733213430754189262, −1.70920117002487773064211473052, −1.70288645393469267938878482188, −0.37378416786152686106688979303,
0.37378416786152686106688979303, 1.70288645393469267938878482188, 1.70920117002487773064211473052, 3.11034124194733213430754189262, 3.37406758071836814924915307209, 3.55890901417255229017515382962, 4.11186049711831909206671923886, 4.98180200108526678902327097794, 5.05015336255964301849953312111, 5.63199499408319090999921245342, 5.80885774896028068653041968795, 6.40454443889734962686846198108, 6.61474828444259376711934654195, 7.49262597290169564919552252457, 7.52596873001703172416476315627, 8.206743980650072918043135650299, 8.392065745180742976954026678755, 8.955466640314035603967422821677, 9.075030076268185221531080378088, 9.850772558168846196110030715334
