| L(s) = 1 | − 4·3-s + 10·9-s + 2·11-s + 2·13-s + 16-s − 4·19-s − 20·27-s − 2·29-s − 8·33-s + 4·37-s − 8·39-s − 2·41-s + 2·47-s − 4·48-s − 4·53-s + 16·57-s + 4·71-s + 4·73-s + 2·79-s + 35·81-s + 2·83-s + 8·87-s + 4·89-s + 20·99-s − 16·111-s − 4·113-s + 20·117-s + ⋯ |
| L(s) = 1 | − 4·3-s + 10·9-s + 2·11-s + 2·13-s + 16-s − 4·19-s − 20·27-s − 2·29-s − 8·33-s + 4·37-s − 8·39-s − 2·41-s + 2·47-s − 4·48-s − 4·53-s + 16·57-s + 4·71-s + 4·73-s + 2·79-s + 35·81-s + 2·83-s + 8·87-s + 4·89-s + 20·99-s − 16·111-s − 4·113-s + 20·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4192644656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4192644656\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.37860111696882272943935716871, −6.27178089563315163664028425376, −6.07801557565012339760402147131, −6.04125062300452721235416149510, −5.75871335249062348630871336349, −5.25638626374358938827516945752, −5.14988178273420134382604218256, −5.08858830245255673536930640089, −4.99521539836493012706220854013, −4.58915883096069873916596771733, −4.30076231456971782866763306704, −4.16985925833762368292470743684, −4.03329527102991937547660542722, −3.84899247429965359862985175003, −3.59097158342244755556324677901, −3.58328680220920211981000917737, −3.33878965528293960305401460200, −2.27797494818232771576332496444, −2.24060520681680874517357841782, −2.00693177149277525959063363460, −1.91019180928855719358095150549, −1.21463547169001404971004794987, −1.12622466143218779620103461745, −1.04671269292018965834359191463, −0.46716688699010155285830984802,
0.46716688699010155285830984802, 1.04671269292018965834359191463, 1.12622466143218779620103461745, 1.21463547169001404971004794987, 1.91019180928855719358095150549, 2.00693177149277525959063363460, 2.24060520681680874517357841782, 2.27797494818232771576332496444, 3.33878965528293960305401460200, 3.58328680220920211981000917737, 3.59097158342244755556324677901, 3.84899247429965359862985175003, 4.03329527102991937547660542722, 4.16985925833762368292470743684, 4.30076231456971782866763306704, 4.58915883096069873916596771733, 4.99521539836493012706220854013, 5.08858830245255673536930640089, 5.14988178273420134382604218256, 5.25638626374358938827516945752, 5.75871335249062348630871336349, 6.04125062300452721235416149510, 6.07801557565012339760402147131, 6.27178089563315163664028425376, 6.37860111696882272943935716871
