| L(s) = 1 | − 2·5-s + 7-s − 4·11-s + 6·13-s − 7·17-s − 5·19-s − 4·23-s + 5·25-s − 2·29-s + 3·31-s − 2·35-s − 11·37-s − 18·41-s + 10·43-s − 3·47-s − 6·49-s − 3·53-s + 8·55-s + 7·59-s − 3·61-s − 12·65-s − 13·67-s − 16·71-s − 7·73-s − 4·77-s + 9·79-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 1.69·17-s − 1.14·19-s − 0.834·23-s + 25-s − 0.371·29-s + 0.538·31-s − 0.338·35-s − 1.80·37-s − 2.81·41-s + 1.52·43-s − 0.437·47-s − 6/7·49-s − 0.412·53-s + 1.07·55-s + 0.911·59-s − 0.384·61-s − 1.48·65-s − 1.58·67-s − 1.89·71-s − 0.819·73-s − 0.455·77-s + 1.01·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−8.543934540086807277270237145478, −8.429330843397724415384928552654, −8.236501128480494015092734713852, −7.87884874230545552450532946852, −7.09640288877772191509778548224, −7.04302612356111764154705568448, −6.46752750286264323400924120464, −6.26142673981552441814272794276, −5.46984948140791200345804814874, −5.43248323373330458389999920710, −4.56050873962319769681726636428, −4.49503671676152370154693441856, −3.98730641678001008285252673794, −3.52745746022227903370012181950, −2.99009812616566836319546321014, −2.52569428731877918263221608383, −1.74424765817236204957901896480, −1.46605579995747644585466527201, 0, 0,
1.46605579995747644585466527201, 1.74424765817236204957901896480, 2.52569428731877918263221608383, 2.99009812616566836319546321014, 3.52745746022227903370012181950, 3.98730641678001008285252673794, 4.49503671676152370154693441856, 4.56050873962319769681726636428, 5.43248323373330458389999920710, 5.46984948140791200345804814874, 6.26142673981552441814272794276, 6.46752750286264323400924120464, 7.04302612356111764154705568448, 7.09640288877772191509778548224, 7.87884874230545552450532946852, 8.236501128480494015092734713852, 8.429330843397724415384928552654, 8.543934540086807277270237145478
