| L(s) = 1 | + 2·7-s − 4·13-s + 2·19-s + 2·31-s − 2·37-s − 4·43-s + 49-s + 2·67-s − 2·73-s − 2·79-s − 8·91-s + 2·103-s + 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2·7-s − 4·13-s + 2·19-s + 2·31-s − 2·37-s − 4·43-s + 49-s + 2·67-s − 2·73-s − 2·79-s − 8·91-s + 2·103-s + 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5999103323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5999103323\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−8.267986284762705842757313145066, −7.76285478212202340210392691558, −7.55917077529776587537660342935, −7.44815481835568423864427563048, −7.27628823452008392492026722041, −6.97827839854047070715630623545, −6.82908323287905615539422532034, −6.60821740181229889229126340408, −6.18462475321189764777263888274, −5.70151990847289172330832249379, −5.43949625678711008288418486788, −5.38915805438657560386696987233, −4.98073153966592754997594122412, −4.84167906884327546955225733171, −4.73803521666916912168467149620, −4.47449960971579833236472073341, −4.33329557414190601727818274402, −3.54170474871652206340908868216, −3.27592709468854609113046103458, −3.00187391093175300179302104255, −2.88972272667109435363341588269, −2.13280923285315758235947114459, −1.96149439826049507956339069820, −1.82436622802639729639815280514, −1.08160545305569271639776234426,
1.08160545305569271639776234426, 1.82436622802639729639815280514, 1.96149439826049507956339069820, 2.13280923285315758235947114459, 2.88972272667109435363341588269, 3.00187391093175300179302104255, 3.27592709468854609113046103458, 3.54170474871652206340908868216, 4.33329557414190601727818274402, 4.47449960971579833236472073341, 4.73803521666916912168467149620, 4.84167906884327546955225733171, 4.98073153966592754997594122412, 5.38915805438657560386696987233, 5.43949625678711008288418486788, 5.70151990847289172330832249379, 6.18462475321189764777263888274, 6.60821740181229889229126340408, 6.82908323287905615539422532034, 6.97827839854047070715630623545, 7.27628823452008392492026722041, 7.44815481835568423864427563048, 7.55917077529776587537660342935, 7.76285478212202340210392691558, 8.267986284762705842757313145066
