| L(s) = 1 | + 5·9-s + 2·11-s + 2·19-s + 12·29-s − 8·31-s − 10·41-s − 49-s − 8·59-s + 12·61-s − 28·71-s + 28·79-s + 16·81-s + 6·89-s + 10·99-s + 20·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 10·171-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 0.603·11-s + 0.458·19-s + 2.22·29-s − 1.43·31-s − 1.56·41-s − 1/7·49-s − 1.04·59-s + 1.53·61-s − 3.32·71-s + 3.15·79-s + 16/9·81-s + 0.635·89-s + 1.00·99-s + 1.91·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.764·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.160380988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.160380988\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.970158244932092815366968472816, −8.578695985536066704380398280713, −8.381849559395472377295807458795, −7.61664346359690389974359169425, −7.50503683514758368259561768628, −7.21302791844666310032264327376, −6.61249687369567945683998830976, −6.41984218277377415154560855052, −6.15893917625170394440255756712, −5.38401529372002869627730417337, −4.96522605835732322251727207776, −4.80289243422002352876560823685, −4.25227263216853171430355368735, −3.72994579584096899222154700627, −3.57536110721281645373913201876, −2.87453571293319934561402060811, −2.34967194082499617238994524512, −1.54493464482122842816710214278, −1.43959317848638518595918552240, −0.60399767384949736125791487850,
0.60399767384949736125791487850, 1.43959317848638518595918552240, 1.54493464482122842816710214278, 2.34967194082499617238994524512, 2.87453571293319934561402060811, 3.57536110721281645373913201876, 3.72994579584096899222154700627, 4.25227263216853171430355368735, 4.80289243422002352876560823685, 4.96522605835732322251727207776, 5.38401529372002869627730417337, 6.15893917625170394440255756712, 6.41984218277377415154560855052, 6.61249687369567945683998830976, 7.21302791844666310032264327376, 7.50503683514758368259561768628, 7.61664346359690389974359169425, 8.381849559395472377295807458795, 8.578695985536066704380398280713, 8.970158244932092815366968472816
