| L(s) = 1 | + 2·3-s − 2·4-s + 2·7-s + 9-s + 7·11-s − 4·12-s − 3·13-s + 4·16-s − 2·17-s − 2·19-s + 4·21-s + 5·23-s − 5·25-s − 4·27-s − 4·28-s + 8·29-s + 14·33-s − 2·36-s − 37-s − 6·39-s + 6·41-s − 4·43-s − 14·44-s + 11·47-s + 8·48-s − 2·49-s − 4·51-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 0.755·7-s + 1/3·9-s + 2.11·11-s − 1.15·12-s − 0.832·13-s + 16-s − 0.485·17-s − 0.458·19-s + 0.872·21-s + 1.04·23-s − 25-s − 0.769·27-s − 0.755·28-s + 1.48·29-s + 2.43·33-s − 1/3·36-s − 0.164·37-s − 0.960·39-s + 0.937·41-s − 0.609·43-s − 2.11·44-s + 1.60·47-s + 1.15·48-s − 2/7·49-s − 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.289630793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.289630793\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11 T + 70 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 50 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 14 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−13.7430110539, −13.3387752108, −12.7567003212, −12.2455260024, −11.9456540192, −11.5304329595, −10.9939490099, −10.3864265066, −9.94269708013, −9.39518751942, −9.06024467999, −8.81456090726, −8.45787316302, −7.77898257149, −7.52766879576, −6.79513331351, −6.32792384251, −5.67267861805, −4.93776087842, −4.47910324158, −4.00116959033, −3.54252664296, −2.70328707502, −1.96124257271, −1.02879498058,
1.02879498058, 1.96124257271, 2.70328707502, 3.54252664296, 4.00116959033, 4.47910324158, 4.93776087842, 5.67267861805, 6.32792384251, 6.79513331351, 7.52766879576, 7.77898257149, 8.45787316302, 8.81456090726, 9.06024467999, 9.39518751942, 9.94269708013, 10.3864265066, 10.9939490099, 11.5304329595, 11.9456540192, 12.2455260024, 12.7567003212, 13.3387752108, 13.7430110539
