| L(s) = 1 | − 2-s − 2·5-s − 2·7-s + 8-s − 3·9-s + 2·10-s − 11-s + 3·13-s + 2·14-s − 16-s + 3·18-s − 16·19-s + 22-s − 7·23-s + 5·25-s − 3·26-s + 7·29-s − 8·31-s + 4·35-s − 2·37-s + 16·38-s − 2·40-s − 6·41-s − 10·43-s + 6·45-s + 7·46-s − 10·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s − 0.755·7-s + 0.353·8-s − 9-s + 0.632·10-s − 0.301·11-s + 0.832·13-s + 0.534·14-s − 1/4·16-s + 0.707·18-s − 3.67·19-s + 0.213·22-s − 1.45·23-s + 25-s − 0.588·26-s + 1.29·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s + 2.59·38-s − 0.316·40-s − 0.937·41-s − 1.52·43-s + 0.894·45-s + 1.03·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 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
−10.33486678994896939404695406822, −10.09779764220304640536630844915, −9.235123960006149298690263809729, −8.799607582241275382596292569310, −8.594077701552727540841591160005, −8.225293853794803904556272665085, −8.060962312770890478669596199577, −7.18354393728661976988611256407, −6.70143131821616749936801186121, −6.47743670333354329450480802341, −5.87376596092880762283267427570, −5.44441939522197989523954849628, −4.54767111561715662528574393116, −4.17791447945268758222376453108, −3.73980658667545622476448159532, −3.05144320309071390402950304270, −2.38715174531484495801190538120, −1.64982940875971812060882349349, 0, 0,
1.64982940875971812060882349349, 2.38715174531484495801190538120, 3.05144320309071390402950304270, 3.73980658667545622476448159532, 4.17791447945268758222376453108, 4.54767111561715662528574393116, 5.44441939522197989523954849628, 5.87376596092880762283267427570, 6.47743670333354329450480802341, 6.70143131821616749936801186121, 7.18354393728661976988611256407, 8.060962312770890478669596199577, 8.225293853794803904556272665085, 8.594077701552727540841591160005, 8.799607582241275382596292569310, 9.235123960006149298690263809729, 10.09779764220304640536630844915, 10.33486678994896939404695406822
