| L(s) = 1 | + 2·5-s − 4·11-s + 2·13-s + 4·17-s − 8·19-s + 8·23-s + 5·25-s − 6·29-s − 8·31-s + 12·37-s + 6·41-s − 4·43-s + 7·49-s − 4·53-s − 8·55-s − 4·59-s + 2·61-s + 4·65-s + 4·67-s + 16·71-s + 20·73-s + 8·79-s + 4·83-s + 8·85-s − 12·89-s − 16·95-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 0.554·13-s + 0.970·17-s − 1.83·19-s + 1.66·23-s + 25-s − 1.11·29-s − 1.43·31-s + 1.97·37-s + 0.937·41-s − 0.609·43-s + 49-s − 0.549·53-s − 1.07·55-s − 0.520·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + 1.89·71-s + 2.34·73-s + 0.900·79-s + 0.439·83-s + 0.867·85-s − 1.27·89-s − 1.64·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066915439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066915439\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.82910976815498650411108649147, −10.55148577808006058343086342745, −9.800694521284593924296395428975, −9.360380108516642797883713940191, −9.296350878961516544050763838765, −8.624007881627953683019060061162, −7.980359729402398523721405884329, −7.959977925105450543168745352543, −7.20850623489491585416891420720, −6.69299185586611711619627501552, −6.34008160466344299310352664117, −5.69395885640079870006386771579, −5.35184931790300732064915431403, −5.02244930777681257047983776151, −4.21186959740816363360918420146, −3.71741783398813422121774248189, −2.93723664021228436927822728302, −2.43903165434531482004299607705, −1.81482394597607346601195346320, −0.795445496484644195298392014606,
0.795445496484644195298392014606, 1.81482394597607346601195346320, 2.43903165434531482004299607705, 2.93723664021228436927822728302, 3.71741783398813422121774248189, 4.21186959740816363360918420146, 5.02244930777681257047983776151, 5.35184931790300732064915431403, 5.69395885640079870006386771579, 6.34008160466344299310352664117, 6.69299185586611711619627501552, 7.20850623489491585416891420720, 7.959977925105450543168745352543, 7.980359729402398523721405884329, 8.624007881627953683019060061162, 9.296350878961516544050763838765, 9.360380108516642797883713940191, 9.800694521284593924296395428975, 10.55148577808006058343086342745, 10.82910976815498650411108649147
