| L(s) = 1 | − 8·4-s − 9·5-s − 18·9-s − 6·11-s + 48·16-s − 38·19-s + 72·20-s + 56·25-s + 144·36-s + 48·44-s + 162·45-s − 73·49-s + 54·55-s − 206·61-s − 256·64-s + 304·76-s − 432·80-s + 243·81-s + 342·95-s + 108·99-s − 448·100-s + 204·101-s − 215·121-s − 279·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9/5·5-s − 2·9-s − 0.545·11-s + 3·16-s − 2·19-s + 18/5·20-s + 2.23·25-s + 4·36-s + 1.09·44-s + 18/5·45-s − 1.48·49-s + 0.981·55-s − 3.37·61-s − 4·64-s + 4·76-s − 5.39·80-s + 3·81-s + 18/5·95-s + 1.09·99-s − 4.47·100-s + 2.01·101-s − 1.77·121-s − 2.23·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07744340896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07744340896\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + 9 T + p^{2} T^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )( 1 + 5 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )( 1 + 15 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 85 T + p^{2} T^{2} )( 1 + 85 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 75 T + p^{2} T^{2} )( 1 + 75 T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 103 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )( 1 + 25 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 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
−14.25157432388585649567356766633, −13.39187719624353869399446367513, −13.01776673187754855297668165522, −12.39799822921906044180539089534, −12.05242044382348750409410876140, −11.37248360954775766794190211647, −10.75816691137317208333703983399, −10.43740418812664159808126451517, −9.366359671506851662218323206288, −8.826988218231914666856229004933, −8.578296919043166736706408665356, −7.87179491560391212294171419485, −7.85984485776406111952369264797, −6.43275358082324250978925055312, −5.72380528146169868913489511550, −4.84872842311756052007724712634, −4.48375704412604770621758707447, −3.63104032948032955842137086463, −3.00563147960642771695247642133, −0.21694977185179685344681100031,
0.21694977185179685344681100031, 3.00563147960642771695247642133, 3.63104032948032955842137086463, 4.48375704412604770621758707447, 4.84872842311756052007724712634, 5.72380528146169868913489511550, 6.43275358082324250978925055312, 7.85984485776406111952369264797, 7.87179491560391212294171419485, 8.578296919043166736706408665356, 8.826988218231914666856229004933, 9.366359671506851662218323206288, 10.43740418812664159808126451517, 10.75816691137317208333703983399, 11.37248360954775766794190211647, 12.05242044382348750409410876140, 12.39799822921906044180539089534, 13.01776673187754855297668165522, 13.39187719624353869399446367513, 14.25157432388585649567356766633
