| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 2·5-s − 4·6-s − 4·8-s + 3·9-s + 4·10-s − 2·11-s + 4·12-s − 8·13-s − 4·15-s + 8·16-s − 10·17-s − 6·18-s − 2·19-s − 4·20-s + 4·22-s − 6·23-s − 8·24-s + 3·25-s + 16·26-s + 4·27-s + 2·29-s + 8·30-s − 6·31-s − 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s + 1.15·12-s − 2.21·13-s − 1.03·15-s + 2·16-s − 2.42·17-s − 1.41·18-s − 0.458·19-s − 0.894·20-s + 0.852·22-s − 1.25·23-s − 1.63·24-s + 3/5·25-s + 3.13·26-s + 0.769·27-s + 0.371·29-s + 1.46·30-s − 1.07·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.849543164466496226958806702435, −9.766898866858391793446602146549, −9.149077310998762148648251151922, −8.906914209968855158380828581096, −8.494829620778494367034299481028, −8.199555656809790592785142199134, −7.55163285068307729115697102553, −7.50428454799348286743511579733, −6.98658786520566803110093170981, −6.40365711689107019910243102145, −5.96228955158126698577274887989, −4.98484599268808484793933805127, −4.60593260342530334305673609658, −4.13747278538977431008876417058, −3.37003269211124588740427460811, −2.73395600229801674050154264895, −2.38783044957083797331845206260, −1.77352868017345184102614907117, 0, 0,
1.77352868017345184102614907117, 2.38783044957083797331845206260, 2.73395600229801674050154264895, 3.37003269211124588740427460811, 4.13747278538977431008876417058, 4.60593260342530334305673609658, 4.98484599268808484793933805127, 5.96228955158126698577274887989, 6.40365711689107019910243102145, 6.98658786520566803110093170981, 7.50428454799348286743511579733, 7.55163285068307729115697102553, 8.199555656809790592785142199134, 8.494829620778494367034299481028, 8.906914209968855158380828581096, 9.149077310998762148648251151922, 9.766898866858391793446602146549, 9.849543164466496226958806702435
