| L(s) = 1 | − 4·2-s + 4·4-s + 16·8-s + 10·9-s − 64·16-s − 40·18-s + 20·23-s − 22·25-s + 64·32-s + 40·36-s − 80·46-s + 88·50-s + 192·64-s − 440·71-s + 160·72-s − 260·79-s + 81·81-s + 80·92-s − 88·100-s − 104·113-s − 242·121-s + 127-s − 768·128-s + 131-s + 137-s + 139-s + 1.76e3·142-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·8-s + 10/9·9-s − 4·16-s − 2.22·18-s + 0.869·23-s − 0.879·25-s + 2·32-s + 10/9·36-s − 1.73·46-s + 1.75·50-s + 3·64-s − 6.19·71-s + 20/9·72-s − 3.29·79-s + 81-s + 0.869·92-s − 0.879·100-s − 0.920·113-s − 2·121-s + 0.00787·127-s − 6·128-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 12.3·142-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1901583247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1901583247\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 10 T^{2} + 19 T^{4} - 10 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^3$ | \( 1 + 22 T^{2} - 141 T^{4} + 22 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 310 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 650 T^{2} + 292179 T^{4} - 650 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T - 429 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 1130 T^{2} - 10840461 T^{4} - 1130 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 7370 T^{2} + 40471059 T^{4} - 7370 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 110 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 130 T + 10659 T^{2} + 130 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13130 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.907613716065811827829959742803, −7.70798413277686910727223293120, −7.70488946817470115297522595834, −7.27519429898870777025891565309, −7.22357037660749062789783740042, −6.95844419856996022186915442009, −6.50561855663540992989160781856, −6.44074727425453942963670884192, −6.05706233022637177551826580116, −5.51005653435672483105474544738, −5.50101705135615644796188014257, −5.03602145480728480289677283295, −4.91009764036652926695445416971, −4.33873156400996331400541059434, −4.26156110973976428261130940309, −3.99235288356955229742321431178, −3.92391334692232488084933112347, −3.11653345360547991509085994362, −2.75789499382040349599541684444, −2.66596688348322788853646395483, −1.69607128792641034346872345437, −1.54219579443392315373091080358, −1.51440646260241593600473976929, −0.817049253821624342113664848996, −0.16626762367564172278337454049,
0.16626762367564172278337454049, 0.817049253821624342113664848996, 1.51440646260241593600473976929, 1.54219579443392315373091080358, 1.69607128792641034346872345437, 2.66596688348322788853646395483, 2.75789499382040349599541684444, 3.11653345360547991509085994362, 3.92391334692232488084933112347, 3.99235288356955229742321431178, 4.26156110973976428261130940309, 4.33873156400996331400541059434, 4.91009764036652926695445416971, 5.03602145480728480289677283295, 5.50101705135615644796188014257, 5.51005653435672483105474544738, 6.05706233022637177551826580116, 6.44074727425453942963670884192, 6.50561855663540992989160781856, 6.95844419856996022186915442009, 7.22357037660749062789783740042, 7.27519429898870777025891565309, 7.70488946817470115297522595834, 7.70798413277686910727223293120, 7.907613716065811827829959742803
