| L(s) = 1 | + 2-s + 3-s + 4·5-s + 6-s − 4·7-s − 8-s + 4·10-s + 11-s − 2·13-s − 4·14-s + 4·15-s − 16-s + 4·17-s + 19-s − 4·21-s + 22-s − 7·23-s − 24-s + 2·25-s − 2·26-s − 27-s + 5·29-s + 4·30-s + 33-s + 4·34-s − 16·35-s + 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1.78·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1.26·10-s + 0.301·11-s − 0.554·13-s − 1.06·14-s + 1.03·15-s − 1/4·16-s + 0.970·17-s + 0.229·19-s − 0.872·21-s + 0.213·22-s − 1.45·23-s − 0.204·24-s + 2/5·25-s − 0.392·26-s − 0.192·27-s + 0.928·29-s + 0.730·30-s + 0.174·33-s + 0.685·34-s − 2.70·35-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736164 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736164 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.784742750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.784742750\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.08911014973339427244333680243, −10.04818742161258756308076726857, −9.552783616905676095264593883122, −9.296260694910650181766154255414, −8.989677057407195517768409327417, −8.105599757107031732519836154586, −8.029694665110982687973210139920, −7.28294326136919536273739058819, −6.77825001292884170708974427047, −6.19290639976824970502718514526, −6.12120465816892413948084301070, −5.62265552974289319933749723644, −5.29388391176442168375027303412, −4.46816339439827326408126456538, −4.04660651529278973277082613165, −3.32994398368226271789861976640, −3.06776895676678392130366285217, −2.23439274028015595111508845256, −2.07956804913000012759231554025, −0.808238895048292322350042728810,
0.808238895048292322350042728810, 2.07956804913000012759231554025, 2.23439274028015595111508845256, 3.06776895676678392130366285217, 3.32994398368226271789861976640, 4.04660651529278973277082613165, 4.46816339439827326408126456538, 5.29388391176442168375027303412, 5.62265552974289319933749723644, 6.12120465816892413948084301070, 6.19290639976824970502718514526, 6.77825001292884170708974427047, 7.28294326136919536273739058819, 8.029694665110982687973210139920, 8.105599757107031732519836154586, 8.989677057407195517768409327417, 9.296260694910650181766154255414, 9.552783616905676095264593883122, 10.04818742161258756308076726857, 10.08911014973339427244333680243
