| L(s) = 1 | + 2·3-s + 4·7-s + 2·9-s − 2·11-s − 2·13-s + 6·19-s + 8·21-s + 12·23-s + 6·27-s − 6·29-s + 16·31-s − 4·33-s + 6·37-s − 4·39-s + 10·43-s − 2·49-s + 10·53-s + 12·57-s − 6·59-s − 18·61-s + 8·63-s − 10·67-s + 24·69-s + 8·73-s − 8·77-s + 11·81-s + 2·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 2/3·9-s − 0.603·11-s − 0.554·13-s + 1.37·19-s + 1.74·21-s + 2.50·23-s + 1.15·27-s − 1.11·29-s + 2.87·31-s − 0.696·33-s + 0.986·37-s − 0.640·39-s + 1.52·43-s − 2/7·49-s + 1.37·53-s + 1.58·57-s − 0.781·59-s − 2.30·61-s + 1.00·63-s − 1.22·67-s + 2.88·69-s + 0.936·73-s − 0.911·77-s + 11/9·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.368365887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.368365887\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.297236129656082950219034357114, −9.271574097510806860633261607808, −8.761414318824999894348447805031, −8.460311740436829434519280396047, −7.934266578093900937375458335451, −7.70344864331412648208238040484, −7.34469432582924521459797946217, −7.17609075176525688818437269853, −6.25569424794157170047928943777, −6.11345292999740449245689439992, −5.11871327533801936140181988879, −5.11755661289014142830855737583, −4.55024001762475265967683469123, −4.41800258016212945509248578004, −3.35078832512169742578649991236, −3.13812131818758889137264684957, −2.57684714268305765122376662778, −2.27531674827328281987956498482, −1.23374554061319535138905960430, −1.02201724135173927526339978237,
1.02201724135173927526339978237, 1.23374554061319535138905960430, 2.27531674827328281987956498482, 2.57684714268305765122376662778, 3.13812131818758889137264684957, 3.35078832512169742578649991236, 4.41800258016212945509248578004, 4.55024001762475265967683469123, 5.11755661289014142830855737583, 5.11871327533801936140181988879, 6.11345292999740449245689439992, 6.25569424794157170047928943777, 7.17609075176525688818437269853, 7.34469432582924521459797946217, 7.70344864331412648208238040484, 7.934266578093900937375458335451, 8.460311740436829434519280396047, 8.761414318824999894348447805031, 9.271574097510806860633261607808, 9.297236129656082950219034357114
