| L(s) = 1 | − 2·3-s − 4·5-s + 3·9-s + 8·15-s − 2·17-s + 2·19-s + 8·23-s + 5·25-s − 10·27-s + 4·29-s − 4·31-s + 6·37-s + 4·41-s − 16·43-s − 12·45-s + 4·47-s + 4·51-s + 10·53-s − 4·57-s − 6·59-s + 4·61-s − 12·67-s − 16·69-s − 14·73-s − 10·75-s − 8·79-s + 20·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s + 2.06·15-s − 0.485·17-s + 0.458·19-s + 1.66·23-s + 25-s − 1.92·27-s + 0.742·29-s − 0.718·31-s + 0.986·37-s + 0.624·41-s − 2.43·43-s − 1.78·45-s + 0.583·47-s + 0.560·51-s + 1.37·53-s − 0.529·57-s − 0.781·59-s + 0.512·61-s − 1.46·67-s − 1.92·69-s − 1.63·73-s − 1.15·75-s − 0.900·79-s + 20/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5135658534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5135658534\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.51626774879680361853368289425, −10.39634671333918661667202689782, −9.670811966317645681185918411209, −9.162424098480658488920603158947, −8.910297209910759403257150780537, −8.159118395749300509145893752973, −7.959165032623729722878548868736, −7.24167264687526580812488089890, −7.23279691262199210163764178735, −6.73584687160972200688912488859, −6.00757487158782994957037193588, −5.72014630663450837996309757767, −5.04246744718464189925255466788, −4.55496294889768908866852938258, −4.34731053421375344325537493103, −3.50346446316351625518264873692, −3.34226234510873272797157465887, −2.32847666281513009834910693752, −1.30355356123409053464692514177, −0.42843877870386109503346527148,
0.42843877870386109503346527148, 1.30355356123409053464692514177, 2.32847666281513009834910693752, 3.34226234510873272797157465887, 3.50346446316351625518264873692, 4.34731053421375344325537493103, 4.55496294889768908866852938258, 5.04246744718464189925255466788, 5.72014630663450837996309757767, 6.00757487158782994957037193588, 6.73584687160972200688912488859, 7.23279691262199210163764178735, 7.24167264687526580812488089890, 7.959165032623729722878548868736, 8.159118395749300509145893752973, 8.910297209910759403257150780537, 9.162424098480658488920603158947, 9.670811966317645681185918411209, 10.39634671333918661667202689782, 10.51626774879680361853368289425
