| L(s) = 1 | − 2-s + 2·7-s + 8-s − 3·11-s + 2·13-s − 2·14-s − 16-s − 6·17-s − 2·19-s + 3·22-s + 6·23-s − 2·26-s + 6·29-s + 4·31-s + 6·34-s + 8·37-s + 2·38-s + 9·41-s − 43-s − 6·46-s + 6·47-s + 7·49-s + 24·53-s + 2·56-s − 6·58-s + 3·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.755·7-s + 0.353·8-s − 0.904·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.639·22-s + 1.25·23-s − 0.392·26-s + 1.11·29-s + 0.718·31-s + 1.02·34-s + 1.31·37-s + 0.324·38-s + 1.40·41-s − 0.152·43-s − 0.884·46-s + 0.875·47-s + 49-s + 3.29·53-s + 0.267·56-s − 0.787·58-s + 0.390·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.568759163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568759163\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−9.722369609237688083788178027478, −9.313283427378480914891551052726, −8.913795991715564887860433611349, −8.624836346510924092251675380814, −8.183829254254717621803134805257, −8.081561546075147392540413369623, −7.33011532617374024752799422165, −7.08301324798306128269987872135, −6.72262884879710443184376247423, −6.04464094972397484768196270859, −5.72012198894698931123076456277, −5.17373671132583908878810495237, −4.69612443582370998233432361168, −4.18816821273531339135060423035, −4.07120939574563063305026107230, −2.99486303820184480794726220337, −2.48622720568250836461044729553, −2.22372656523714028747548113567, −1.15937606750156101525481152371, −0.68202309208434869562490035590,
0.68202309208434869562490035590, 1.15937606750156101525481152371, 2.22372656523714028747548113567, 2.48622720568250836461044729553, 2.99486303820184480794726220337, 4.07120939574563063305026107230, 4.18816821273531339135060423035, 4.69612443582370998233432361168, 5.17373671132583908878810495237, 5.72012198894698931123076456277, 6.04464094972397484768196270859, 6.72262884879710443184376247423, 7.08301324798306128269987872135, 7.33011532617374024752799422165, 8.081561546075147392540413369623, 8.183829254254717621803134805257, 8.624836346510924092251675380814, 8.913795991715564887860433611349, 9.313283427378480914891551052726, 9.722369609237688083788178027478
