| L(s) = 1 | + 3-s + 4-s + 2·7-s + 9-s + 12-s + 16-s − 2·19-s + 2·21-s − 9·25-s + 27-s + 2·28-s + 8·31-s + 36-s + 6·37-s + 18·43-s + 48-s − 11·49-s − 2·57-s + 12·61-s + 2·63-s + 64-s + 24·67-s − 20·73-s − 9·75-s − 2·76-s + 20·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1/4·16-s − 0.458·19-s + 0.436·21-s − 9/5·25-s + 0.192·27-s + 0.377·28-s + 1.43·31-s + 1/6·36-s + 0.986·37-s + 2.74·43-s + 0.144·48-s − 1.57·49-s − 0.264·57-s + 1.53·61-s + 0.251·63-s + 1/8·64-s + 2.93·67-s − 2.34·73-s − 1.03·75-s − 0.229·76-s + 2.25·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90828 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90828 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287833704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287833704\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−9.808192641909838259670499406456, −9.212591262870804148890833263566, −8.394490307116038147042717026528, −8.240694861078888273486995722554, −7.63591160976614872787897721858, −7.39230013101638678138560124593, −6.35948215250638977727528832831, −6.34044350417483302017587248899, −5.44738455570614240033325393245, −4.88979839840342254384190809437, −4.07632477811469157902271861358, −3.79639747241266837059275722803, −2.60778907267363927767629977323, −2.30356803383598646559796068707, −1.23586403051660851431378504272,
1.23586403051660851431378504272, 2.30356803383598646559796068707, 2.60778907267363927767629977323, 3.79639747241266837059275722803, 4.07632477811469157902271861358, 4.88979839840342254384190809437, 5.44738455570614240033325393245, 6.34044350417483302017587248899, 6.35948215250638977727528832831, 7.39230013101638678138560124593, 7.63591160976614872787897721858, 8.240694861078888273486995722554, 8.394490307116038147042717026528, 9.212591262870804148890833263566, 9.808192641909838259670499406456
