| L(s) = 1 | − 5·9-s − 16·11-s + 16·19-s + 14·29-s + 16·31-s + 10·41-s + 28·49-s + 6·59-s − 22·61-s − 22·71-s − 26·79-s + 9·81-s + 6·89-s + 80·99-s + 2·101-s + 6·109-s + 116·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 4.82·11-s + 3.67·19-s + 2.59·29-s + 2.87·31-s + 1.56·41-s + 4·49-s + 0.781·59-s − 2.81·61-s − 2.61·71-s − 2.92·79-s + 81-s + 0.635·89-s + 8.04·99-s + 0.199·101-s + 0.574·109-s + 10.5·121-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 − 1.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8687610545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8687610545\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 21 T^{2} - 88 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 15 T^{2} - 2584 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 79 T^{2} + 912 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.38356424081060353592215435370, −6.28669796569647541201119120229, −5.92509620072304508060286153715, −5.79643385685210807829096836275, −5.79171647359619709129604771735, −5.39563746090040951447782600225, −5.30772673196891079569223759033, −5.16890585522171654022793949213, −4.93835238718973273443240482952, −4.77992128051291513828435032891, −4.42707019747951498947994734060, −4.23072993316367225933470201142, −4.10763061056455651179022011292, −3.43902332191729874695336194360, −3.11685102632212863325851860133, −2.95621952386825897466200902077, −2.95528333170578977248903306998, −2.65327265003207122259124315320, −2.63789883043029607334265364147, −2.46669442296778826024636223204, −2.01622462587638313738674687362, −1.15314612026434374664625946431, −1.12850176975616747488141956655, −0.74860510570705409614897713984, −0.20949064793891694032468013396,
0.20949064793891694032468013396, 0.74860510570705409614897713984, 1.12850176975616747488141956655, 1.15314612026434374664625946431, 2.01622462587638313738674687362, 2.46669442296778826024636223204, 2.63789883043029607334265364147, 2.65327265003207122259124315320, 2.95528333170578977248903306998, 2.95621952386825897466200902077, 3.11685102632212863325851860133, 3.43902332191729874695336194360, 4.10763061056455651179022011292, 4.23072993316367225933470201142, 4.42707019747951498947994734060, 4.77992128051291513828435032891, 4.93835238718973273443240482952, 5.16890585522171654022793949213, 5.30772673196891079569223759033, 5.39563746090040951447782600225, 5.79171647359619709129604771735, 5.79643385685210807829096836275, 5.92509620072304508060286153715, 6.28669796569647541201119120229, 6.38356424081060353592215435370
