| L(s) = 1 | − 7-s − 4·9-s − 2·11-s + 6·23-s + 2·25-s − 6·29-s + 2·37-s + 6·43-s + 49-s + 8·53-s + 4·63-s − 12·67-s − 12·71-s + 2·77-s + 7·81-s + 8·99-s + 12·107-s + 2·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 6·161-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 4/3·9-s − 0.603·11-s + 1.25·23-s + 2/5·25-s − 1.11·29-s + 0.328·37-s + 0.914·43-s + 1/7·49-s + 1.09·53-s + 0.503·63-s − 1.46·67-s − 1.42·71-s + 0.227·77-s + 7/9·81-s + 0.804·99-s + 1.16·107-s + 0.191·109-s − 1.63·121-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 − 0.472·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
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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
−7.75222761010965240609476780265, −7.33171883339341772994779710092, −6.89009920778461053670689200555, −6.39988256335822680138730366270, −5.87962956974246304838943353175, −5.52745844147401835353288767809, −5.26577017746525057508095490793, −4.56766962697825713558786982931, −4.14401623156428438521805723751, −3.40797472777902592444016143311, −2.97791501497783302352770740114, −2.63975336852309235676702520365, −1.93019087833099441846701632153, −0.938184998588665453621265998161, 0,
0.938184998588665453621265998161, 1.93019087833099441846701632153, 2.63975336852309235676702520365, 2.97791501497783302352770740114, 3.40797472777902592444016143311, 4.14401623156428438521805723751, 4.56766962697825713558786982931, 5.26577017746525057508095490793, 5.52745844147401835353288767809, 5.87962956974246304838943353175, 6.39988256335822680138730366270, 6.89009920778461053670689200555, 7.33171883339341772994779710092, 7.75222761010965240609476780265
