| L(s) = 1 | − 7-s + 4·13-s + 19-s − 25-s + 31-s − 2·37-s − 2·43-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s − 2·97-s − 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | − 7-s + 4·13-s + 19-s − 25-s + 31-s − 2·37-s − 2·43-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s − 2·97-s − 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9457631337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9457631337\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−10.62932871528426230517026056625, −10.50220222348165524604662537221, −9.742232412987617180579576569954, −9.687532573179586339968001977749, −8.870239416831465612952893521537, −8.699164035819916900282065218251, −8.232794775295628069526212849968, −8.034532254963809259935478754537, −7.10171719857339533757056547881, −6.78871695268480433648327619841, −6.34137702286563838020390891255, −5.96747182206304400887987792285, −5.61043948651723752723535650071, −5.04877944921169936566024468690, −4.00545553845051295599186923696, −3.92259897083775955625535401179, −3.18449586044372927020330116682, −3.10468264816957414902063507855, −1.68767077795690034784893235093, −1.26918191813805124151530000226,
1.26918191813805124151530000226, 1.68767077795690034784893235093, 3.10468264816957414902063507855, 3.18449586044372927020330116682, 3.92259897083775955625535401179, 4.00545553845051295599186923696, 5.04877944921169936566024468690, 5.61043948651723752723535650071, 5.96747182206304400887987792285, 6.34137702286563838020390891255, 6.78871695268480433648327619841, 7.10171719857339533757056547881, 8.034532254963809259935478754537, 8.232794775295628069526212849968, 8.699164035819916900282065218251, 8.870239416831465612952893521537, 9.687532573179586339968001977749, 9.742232412987617180579576569954, 10.50220222348165524604662537221, 10.62932871528426230517026056625
