| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s + 2·5-s − 4·6-s + 2·9-s − 4·10-s − 6·11-s + 4·12-s + 6·13-s + 4·15-s − 4·16-s − 4·18-s − 2·19-s + 4·20-s + 12·22-s − 16·23-s − 25-s − 12·26-s + 6·27-s + 6·29-s − 8·30-s + 8·32-s − 12·33-s + 4·36-s + 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s + 2/3·9-s − 1.26·10-s − 1.80·11-s + 1.15·12-s + 1.66·13-s + 1.03·15-s − 16-s − 0.942·18-s − 0.458·19-s + 0.894·20-s + 2.55·22-s − 3.33·23-s − 1/5·25-s − 2.35·26-s + 1.15·27-s + 1.11·29-s − 1.46·30-s + 1.41·32-s − 2.08·33-s + 2/3·36-s + 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6992058157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6992058157\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−14.40961390949872677282428971982, −14.16526480917337796251134109767, −13.53370766369937713346205913049, −13.35575430209823148734361938811, −12.65503725385643908561619473780, −11.90748346611457144115278497932, −10.88115227422887522374158348317, −10.69096341676788920220445688197, −9.977915016614731615484456961755, −9.731379494354792100923490262160, −9.047040914803898684021525532806, −8.283214005580134722423170567174, −7.955042095620064103982772188440, −7.88841615959089504205364663306, −6.30026986621631158380811466648, −6.25801948258685009188092670275, −4.94519126799145894471468277245, −3.87649783842984427497926954231, −2.63230194781690904930246602242, −1.87808949039494321075128461614,
1.87808949039494321075128461614, 2.63230194781690904930246602242, 3.87649783842984427497926954231, 4.94519126799145894471468277245, 6.25801948258685009188092670275, 6.30026986621631158380811466648, 7.88841615959089504205364663306, 7.955042095620064103982772188440, 8.283214005580134722423170567174, 9.047040914803898684021525532806, 9.731379494354792100923490262160, 9.977915016614731615484456961755, 10.69096341676788920220445688197, 10.88115227422887522374158348317, 11.90748346611457144115278497932, 12.65503725385643908561619473780, 13.35575430209823148734361938811, 13.53370766369937713346205913049, 14.16526480917337796251134109767, 14.40961390949872677282428971982
