| L(s) = 1 | + 3-s − 3·5-s + 5·7-s + 9·11-s − 3·15-s + 6·17-s − 2·19-s + 5·21-s + 12·23-s + 25-s − 27-s − 18·29-s − 31-s + 9·33-s − 15·35-s + 2·37-s + 18·49-s + 6·51-s − 9·53-s − 27·55-s − 2·57-s + 3·59-s − 12·61-s + 12·69-s + 12·73-s + 75-s + 45·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.88·7-s + 2.71·11-s − 0.774·15-s + 1.45·17-s − 0.458·19-s + 1.09·21-s + 2.50·23-s + 1/5·25-s − 0.192·27-s − 3.34·29-s − 0.179·31-s + 1.56·33-s − 2.53·35-s + 0.328·37-s + 18/7·49-s + 0.840·51-s − 1.23·53-s − 3.64·55-s − 0.264·57-s + 0.390·59-s − 1.53·61-s + 1.44·69-s + 1.40·73-s + 0.115·75-s + 5.12·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.239584989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.239584989\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 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
−11.54187926698352990356291130730, −11.40187760642489256859797644727, −11.06923139706759261001049053294, −10.77579318983920408378467947379, −9.576448615705121078798692990297, −9.379765717289637018355223749564, −9.018740275008013666836687042112, −8.407096918039041305962134391410, −8.033953770188929567152140076889, −7.63966042431215133832644009409, −6.96357163566334917847979606178, −6.95291130392649731373934382693, −5.57950357124466963762755562911, −5.56732304041548647676529635250, −4.39810343445330545749647737313, −4.23913145348770025646154115123, −3.64691815151268698760953616526, −3.08895961923263020741200153793, −1.67629110394513976347941053883, −1.31041115287137928192876313676,
1.31041115287137928192876313676, 1.67629110394513976347941053883, 3.08895961923263020741200153793, 3.64691815151268698760953616526, 4.23913145348770025646154115123, 4.39810343445330545749647737313, 5.56732304041548647676529635250, 5.57950357124466963762755562911, 6.95291130392649731373934382693, 6.96357163566334917847979606178, 7.63966042431215133832644009409, 8.033953770188929567152140076889, 8.407096918039041305962134391410, 9.018740275008013666836687042112, 9.379765717289637018355223749564, 9.576448615705121078798692990297, 10.77579318983920408378467947379, 11.06923139706759261001049053294, 11.40187760642489256859797644727, 11.54187926698352990356291130730
