| L(s) = 1 | + 5-s − 4·11-s − 6·13-s + 12·17-s − 8·19-s − 2·29-s + 8·31-s − 4·37-s − 6·41-s − 12·43-s + 8·47-s + 7·49-s − 12·53-s − 4·55-s + 12·59-s − 14·61-s − 6·65-s − 4·67-s − 16·71-s − 12·73-s + 8·79-s − 12·83-s + 12·85-s − 20·89-s − 8·95-s − 2·97-s + 6·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 1.66·13-s + 2.91·17-s − 1.83·19-s − 0.371·29-s + 1.43·31-s − 0.657·37-s − 0.937·41-s − 1.82·43-s + 1.16·47-s + 49-s − 1.64·53-s − 0.539·55-s + 1.56·59-s − 1.79·61-s − 0.744·65-s − 0.488·67-s − 1.89·71-s − 1.40·73-s + 0.900·79-s − 1.31·83-s + 1.30·85-s − 2.11·89-s − 0.820·95-s − 0.203·97-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6468931247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6468931247\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 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 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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
−8.910209269918978888702986569809, −8.322873086338119079293919003831, −8.045604733230375329004763712853, −7.76757461116773140042942922589, −7.48022522321709425468438581927, −6.85093745763670635652888586003, −6.81364482004581152325026080271, −6.06288676651208876461685178181, −5.77736851132505997741793556810, −5.32497155985260880778888837747, −5.22566176331012922434777711248, −4.53925202465065590897985155129, −4.40213612816041739260271274480, −3.64814705817298084049408065344, −3.16836673561107486553005468021, −2.62052270679392534567657702583, −2.60915093873376136705339227485, −1.65092101341958065858140906244, −1.39097051699525463643328629306, −0.23843500139855372075214458488,
0.23843500139855372075214458488, 1.39097051699525463643328629306, 1.65092101341958065858140906244, 2.60915093873376136705339227485, 2.62052270679392534567657702583, 3.16836673561107486553005468021, 3.64814705817298084049408065344, 4.40213612816041739260271274480, 4.53925202465065590897985155129, 5.22566176331012922434777711248, 5.32497155985260880778888837747, 5.77736851132505997741793556810, 6.06288676651208876461685178181, 6.81364482004581152325026080271, 6.85093745763670635652888586003, 7.48022522321709425468438581927, 7.76757461116773140042942922589, 8.045604733230375329004763712853, 8.322873086338119079293919003831, 8.910209269918978888702986569809
