| L(s) = 1 | − 2-s + 2·4-s − 2·5-s + 7-s − 5·8-s + 2·10-s + 4·11-s + 2·13-s − 14-s + 5·16-s + 12·17-s + 8·19-s − 4·20-s − 4·22-s + 5·25-s − 2·26-s + 2·28-s − 2·29-s − 10·32-s − 12·34-s − 2·35-s + 12·37-s − 8·38-s + 10·40-s + 2·41-s + 4·43-s + 8·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 0.894·5-s + 0.377·7-s − 1.76·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s − 0.267·14-s + 5/4·16-s + 2.91·17-s + 1.83·19-s − 0.894·20-s − 0.852·22-s + 25-s − 0.392·26-s + 0.377·28-s − 0.371·29-s − 1.76·32-s − 2.05·34-s − 0.338·35-s + 1.97·37-s − 1.29·38-s + 1.58·40-s + 0.312·41-s + 0.609·43-s + 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.623058640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623058640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92450201753723108839159723850, −10.70731250707372895790968129842, −9.858738689022464400491726851907, −9.591201153340732674910822238006, −9.338145666460537515354315811243, −8.828559430282547778411203523050, −8.083027083479950121941719978622, −7.986067018453158391574882202089, −7.43596156329780616969054871519, −7.18563224637505461237775186972, −6.34751108222972485962694119457, −6.16140226457456699023422821152, −5.43133796436705680912532916802, −5.19548638417327909492650328365, −4.05831821280693449248314312925, −3.64760841523237647680286086877, −3.13902494119266651398648008825, −2.67897545623779410770913438795, −1.28787422604450895769164124918, −1.01961052068270479706967825022,
1.01961052068270479706967825022, 1.28787422604450895769164124918, 2.67897545623779410770913438795, 3.13902494119266651398648008825, 3.64760841523237647680286086877, 4.05831821280693449248314312925, 5.19548638417327909492650328365, 5.43133796436705680912532916802, 6.16140226457456699023422821152, 6.34751108222972485962694119457, 7.18563224637505461237775186972, 7.43596156329780616969054871519, 7.986067018453158391574882202089, 8.083027083479950121941719978622, 8.828559430282547778411203523050, 9.338145666460537515354315811243, 9.591201153340732674910822238006, 9.858738689022464400491726851907, 10.70731250707372895790968129842, 10.92450201753723108839159723850
