| L(s) = 1 | + 2-s − 3-s − 6-s + 2·7-s − 8-s − 11-s + 2·13-s + 2·14-s − 16-s + 2·17-s − 6·19-s − 2·21-s − 22-s − 3·23-s + 24-s + 2·26-s + 27-s + 29-s − 6·31-s + 33-s + 2·34-s − 5·37-s − 6·38-s − 2·39-s − 10·41-s − 2·42-s + 5·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 0.301·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 1.37·19-s − 0.436·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.185·29-s − 1.07·31-s + 0.174·33-s + 0.342·34-s − 0.821·37-s − 0.973·38-s − 0.320·39-s − 1.56·41-s − 0.308·42-s + 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.445311879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445311879\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.447948171685922468303356973812, −8.836158745282127657389953706319, −8.606470293009216544830825373937, −8.054037025317572479060976434079, −8.021457343232093909530474749037, −7.41446670446637681567509061195, −6.77106154802300514445689093220, −6.61833685221777775540119662361, −6.08611490894394243747435395065, −5.78141620631240828237904187369, −5.17238315114426841785590913217, −5.09692441843722452212991527479, −4.55132332893147667219621290540, −4.10211022042414312580459619035, −3.61506365155259926654535310376, −3.25645580167666501709840813323, −2.53369776799390139745525541644, −1.85833445375359748484204832258, −1.51907756257393823511908774901, −0.39505957016358572021942899589,
0.39505957016358572021942899589, 1.51907756257393823511908774901, 1.85833445375359748484204832258, 2.53369776799390139745525541644, 3.25645580167666501709840813323, 3.61506365155259926654535310376, 4.10211022042414312580459619035, 4.55132332893147667219621290540, 5.09692441843722452212991527479, 5.17238315114426841785590913217, 5.78141620631240828237904187369, 6.08611490894394243747435395065, 6.61833685221777775540119662361, 6.77106154802300514445689093220, 7.41446670446637681567509061195, 8.021457343232093909530474749037, 8.054037025317572479060976434079, 8.606470293009216544830825373937, 8.836158745282127657389953706319, 9.447948171685922468303356973812
