| L(s) = 1 | + 2·3-s − 6·7-s + 2·9-s − 6·13-s − 2·17-s − 8·19-s − 12·21-s − 2·23-s + 6·27-s + 2·37-s − 12·39-s − 20·41-s + 10·43-s − 6·47-s + 18·49-s − 4·51-s + 10·53-s − 16·57-s + 24·59-s + 4·61-s − 12·63-s − 2·67-s − 4·69-s − 2·73-s − 16·79-s + 11·81-s + 10·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.26·7-s + 2/3·9-s − 1.66·13-s − 0.485·17-s − 1.83·19-s − 2.61·21-s − 0.417·23-s + 1.15·27-s + 0.328·37-s − 1.92·39-s − 3.12·41-s + 1.52·43-s − 0.875·47-s + 18/7·49-s − 0.560·51-s + 1.37·53-s − 2.11·57-s + 3.12·59-s + 0.512·61-s − 1.51·63-s − 0.244·67-s − 0.481·69-s − 0.234·73-s − 1.80·79-s + 11/9·81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9947770355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9947770355\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.19919946969799408617834836787, −10.12019770827379467208749869707, −9.724690470722904152278049104281, −8.991661113158697161383666627446, −8.987052401958675011738244603072, −8.452724393400733958631181553980, −8.069463989498949098821476824004, −7.38569260731394457793209783807, −6.92574740119835516732910322817, −6.63422051205586832692203262238, −6.44874293123435635520929336145, −5.60819060416649640444834770186, −5.17525279931901536021265829349, −4.24383002231607724826439919838, −4.18371058760003549663218260902, −3.26785303674686534517367493068, −3.09577955010136171888452269471, −2.22325946034601879952506452022, −2.21567402034006199968334476966, −0.42870044250894246003153245453,
0.42870044250894246003153245453, 2.21567402034006199968334476966, 2.22325946034601879952506452022, 3.09577955010136171888452269471, 3.26785303674686534517367493068, 4.18371058760003549663218260902, 4.24383002231607724826439919838, 5.17525279931901536021265829349, 5.60819060416649640444834770186, 6.44874293123435635520929336145, 6.63422051205586832692203262238, 6.92574740119835516732910322817, 7.38569260731394457793209783807, 8.069463989498949098821476824004, 8.452724393400733958631181553980, 8.987052401958675011738244603072, 8.991661113158697161383666627446, 9.724690470722904152278049104281, 10.12019770827379467208749869707, 10.19919946969799408617834836787
