| L(s) = 1 | + 2·7-s + 11-s − 6·13-s + 17-s − 2·19-s + 8·23-s − 5·25-s − 5·29-s − 11·31-s + 4·37-s + 9·41-s − 16·43-s + 6·47-s + 3·49-s − 5·53-s + 6·59-s − 8·61-s − 67-s + 8·71-s − 11·73-s + 2·77-s − 8·79-s + 83-s + 2·89-s − 12·91-s − 18·97-s + 6·101-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s − 1.66·13-s + 0.242·17-s − 0.458·19-s + 1.66·23-s − 25-s − 0.928·29-s − 1.97·31-s + 0.657·37-s + 1.40·41-s − 2.43·43-s + 0.875·47-s + 3/7·49-s − 0.686·53-s + 0.781·59-s − 1.02·61-s − 0.122·67-s + 0.949·71-s − 1.28·73-s + 0.227·77-s − 0.900·79-s + 0.109·83-s + 0.211·89-s − 1.25·91-s − 1.82·97-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 11 T + 91 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 91 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 130 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 51 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 13 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 123 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 115 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 154 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 65 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
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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
−7.43542493027224997737351180844, −7.32467690921177015388629561643, −6.90576117012102813410225599228, −6.63673183425818516865636857408, −5.99078286479444574013533264904, −5.89727521308227951996512601245, −5.23570394305607498854212253347, −5.20278970116857737615150407743, −4.85098158804577420470472341087, −4.42707216845791545389721235972, −3.92563260073795340609190950335, −3.80091515961461830913863109897, −3.15872403130141465593431373405, −2.83129301551225469370569455608, −2.19402882379646767640728040003, −2.14919722001601945632705528477, −1.39528374035394068376229403899, −1.16724964842975359833780421932, 0, 0,
1.16724964842975359833780421932, 1.39528374035394068376229403899, 2.14919722001601945632705528477, 2.19402882379646767640728040003, 2.83129301551225469370569455608, 3.15872403130141465593431373405, 3.80091515961461830913863109897, 3.92563260073795340609190950335, 4.42707216845791545389721235972, 4.85098158804577420470472341087, 5.20278970116857737615150407743, 5.23570394305607498854212253347, 5.89727521308227951996512601245, 5.99078286479444574013533264904, 6.63673183425818516865636857408, 6.90576117012102813410225599228, 7.32467690921177015388629561643, 7.43542493027224997737351180844
