| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 3·9-s + 12·11-s + 2·12-s + 16-s + 6·17-s − 3·18-s − 2·19-s − 12·22-s − 2·24-s − 25-s + 4·27-s − 32-s + 24·33-s − 6·34-s + 3·36-s + 2·38-s − 18·41-s − 14·43-s + 12·44-s + 2·48-s + 11·49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 9-s + 3.61·11-s + 0.577·12-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.458·19-s − 2.55·22-s − 0.408·24-s − 1/5·25-s + 0.769·27-s − 0.176·32-s + 4.17·33-s − 1.02·34-s + 1/2·36-s + 0.324·38-s − 2.81·41-s − 2.13·43-s + 1.80·44-s + 0.288·48-s + 11/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968832 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968832 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.312426779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.312426779\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.443424828695416122862075486969, −7.83268146353528315500962133734, −7.12631399441854341066044092332, −6.97231192698558333316844758012, −6.59736023434001425757802101345, −6.16301211985693377316145343724, −5.57702324808249193627299681426, −4.85512283254279582558561488230, −4.21169573330828371132724290124, −3.70435541571971822099748167322, −3.52011913238331407509265098559, −2.96184119851211848322696372329, −1.85413035668575826478952275072, −1.63272357936029339660940403376, −0.999047580680805407462955288675,
0.999047580680805407462955288675, 1.63272357936029339660940403376, 1.85413035668575826478952275072, 2.96184119851211848322696372329, 3.52011913238331407509265098559, 3.70435541571971822099748167322, 4.21169573330828371132724290124, 4.85512283254279582558561488230, 5.57702324808249193627299681426, 6.16301211985693377316145343724, 6.59736023434001425757802101345, 6.97231192698558333316844758012, 7.12631399441854341066044092332, 7.83268146353528315500962133734, 8.443424828695416122862075486969
