| L(s) = 1 | + 7-s − 3·19-s − 25-s − 3·31-s − 37-s + 2·43-s + 67-s + 3·73-s + 79-s + 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 7-s − 3·19-s − 25-s − 3·31-s − 37-s + 2·43-s + 67-s + 3·73-s + 79-s + 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5675397558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5675397558\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−12.53215418346690045431137012082, −12.17999464888387279957474132642, −11.24298370669853202270486110399, −11.20793083446542881049647399051, −10.53267159925723775319435664971, −10.53020636809064090326126324008, −9.356016630938252112091714826295, −9.311494397191903195635700960849, −8.434738428211970939090069030582, −8.344641000529020091597113578714, −7.58550144724325126417600350942, −7.19442353313728619277730897997, −6.43820644479442424950699200760, −5.99967589066269309991929784482, −5.29873520295787314240505170371, −4.78770231110273723801441485478, −3.95066593784166164853909707177, −3.72761380043161358497732490050, −2.18384318450784974480618441631, −1.98126574448115951940535471276,
1.98126574448115951940535471276, 2.18384318450784974480618441631, 3.72761380043161358497732490050, 3.95066593784166164853909707177, 4.78770231110273723801441485478, 5.29873520295787314240505170371, 5.99967589066269309991929784482, 6.43820644479442424950699200760, 7.19442353313728619277730897997, 7.58550144724325126417600350942, 8.344641000529020091597113578714, 8.434738428211970939090069030582, 9.311494397191903195635700960849, 9.356016630938252112091714826295, 10.53020636809064090326126324008, 10.53267159925723775319435664971, 11.20793083446542881049647399051, 11.24298370669853202270486110399, 12.17999464888387279957474132642, 12.53215418346690045431137012082
