| L(s) = 1 | + 2·3-s − 5-s + 7-s + 3·9-s + 2·13-s − 2·15-s − 2·19-s + 2·21-s − 23-s + 25-s + 4·27-s − 35-s + 4·39-s − 3·45-s − 4·57-s − 2·59-s − 61-s + 3·63-s − 2·65-s − 2·69-s + 2·71-s + 2·75-s − 79-s + 5·81-s − 2·83-s + 2·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 2·3-s − 5-s + 7-s + 3·9-s + 2·13-s − 2·15-s − 2·19-s + 2·21-s − 23-s + 25-s + 4·27-s − 35-s + 4·39-s − 3·45-s − 4·57-s − 2·59-s − 61-s + 3·63-s − 2·65-s − 2·69-s + 2·71-s + 2·75-s − 79-s + 5·81-s − 2·83-s + 2·91-s + 2·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.699687925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.699687925\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.235072933848291361251318556089, −9.006182579337892854141571062135, −8.505947450301507527814973318288, −8.455864843244766535547882207481, −7.991306080437785661021347412388, −7.964290299179779329584212328602, −7.36774723319439256077588380081, −6.97423186950906183680348929165, −6.36542483901567264389734633529, −6.28595586232714010796006816735, −5.52146208039271700821980634512, −4.77570975112722388148692413951, −4.32855987043820803936163721988, −4.24096841537220136804928842947, −3.73128407850930277592254016436, −3.35451779897366599851214696418, −2.83066632861473287727777596446, −2.19984469129728737511217979537, −1.66634993513980900622598223082, −1.26294782353213111319871103588,
1.26294782353213111319871103588, 1.66634993513980900622598223082, 2.19984469129728737511217979537, 2.83066632861473287727777596446, 3.35451779897366599851214696418, 3.73128407850930277592254016436, 4.24096841537220136804928842947, 4.32855987043820803936163721988, 4.77570975112722388148692413951, 5.52146208039271700821980634512, 6.28595586232714010796006816735, 6.36542483901567264389734633529, 6.97423186950906183680348929165, 7.36774723319439256077588380081, 7.964290299179779329584212328602, 7.991306080437785661021347412388, 8.455864843244766535547882207481, 8.505947450301507527814973318288, 9.006182579337892854141571062135, 9.235072933848291361251318556089
