| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s + 5·11-s − 16-s + 4·17-s + 8·19-s + 5·22-s − 4·23-s + 5·25-s + 10·29-s + 3·31-s + 4·34-s + 4·37-s + 8·38-s + 40-s + 4·43-s − 4·46-s + 6·47-s + 5·50-s − 9·53-s − 5·55-s + 10·58-s + 11·59-s − 6·61-s + 3·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.50·11-s − 1/4·16-s + 0.970·17-s + 1.83·19-s + 1.06·22-s − 0.834·23-s + 25-s + 1.85·29-s + 0.538·31-s + 0.685·34-s + 0.657·37-s + 1.29·38-s + 0.158·40-s + 0.609·43-s − 0.589·46-s + 0.875·47-s + 0.707·50-s − 1.23·53-s − 0.674·55-s + 1.31·58-s + 1.43·59-s − 0.768·61-s + 0.381·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.341585564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.341585564\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11540704248499395603415540464, −10.01143729166675225316884430746, −9.519006735454344704222507572661, −9.164414113517100350406773077012, −8.624217823461401502501088680952, −8.255500785886550559870812573355, −7.77536487074137727204332385382, −7.35610229703135606974806560249, −6.80217048114000015377237886980, −6.49984525969903431241164747872, −5.91564569081852378850912014239, −5.57694513507731158627788298420, −4.89335366459859135369220866276, −4.61728997165275059376449943336, −3.90094578111485467999594722712, −3.71391614975541884787570827028, −2.95284197462116289950511356495, −2.64662095944537389539110956553, −1.33840474573420498753116503749, −0.947167516877525195914721387891,
0.947167516877525195914721387891, 1.33840474573420498753116503749, 2.64662095944537389539110956553, 2.95284197462116289950511356495, 3.71391614975541884787570827028, 3.90094578111485467999594722712, 4.61728997165275059376449943336, 4.89335366459859135369220866276, 5.57694513507731158627788298420, 5.91564569081852378850912014239, 6.49984525969903431241164747872, 6.80217048114000015377237886980, 7.35610229703135606974806560249, 7.77536487074137727204332385382, 8.255500785886550559870812573355, 8.624217823461401502501088680952, 9.164414113517100350406773077012, 9.519006735454344704222507572661, 10.01143729166675225316884430746, 10.11540704248499395603415540464
