| L(s) = 1 | + 2·5-s + 9-s + 4·11-s − 25-s + 16·31-s + 16·37-s + 2·45-s − 8·47-s − 10·49-s + 8·55-s − 8·59-s − 8·67-s + 81-s + 20·89-s − 24·97-s + 4·99-s + 16·103-s − 8·113-s + 5·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s + 1.20·11-s − 1/5·25-s + 2.87·31-s + 2.63·37-s + 0.298·45-s − 1.16·47-s − 1.42·49-s + 1.07·55-s − 1.04·59-s − 0.977·67-s + 1/9·81-s + 2.11·89-s − 2.43·97-s + 0.402·99-s + 1.57·103-s − 0.752·113-s + 5/11·121-s − 1.07·125-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 + 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.300469852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.300469852\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.016667126632754198717649156669, −7.41775555739278062547456222199, −6.79770818473963076739706443831, −6.41235739359443040865154568096, −6.18628804225171192932001496700, −5.93037231815654555586534506900, −5.15321599540511068580600739648, −4.63905948797132820556735480946, −4.40072186473316933831203819181, −3.87452114942047442611502791181, −3.06595887645954977350650783237, −2.79578390065015919648607782240, −2.00371462393693381120589554650, −1.44670541430155031299058021490, −0.822027003142296897017217460815,
0.822027003142296897017217460815, 1.44670541430155031299058021490, 2.00371462393693381120589554650, 2.79578390065015919648607782240, 3.06595887645954977350650783237, 3.87452114942047442611502791181, 4.40072186473316933831203819181, 4.63905948797132820556735480946, 5.15321599540511068580600739648, 5.93037231815654555586534506900, 6.18628804225171192932001496700, 6.41235739359443040865154568096, 6.79770818473963076739706443831, 7.41775555739278062547456222199, 8.016667126632754198717649156669
