| L(s) = 1 | + 4·2-s + 10·4-s + 2·5-s + 20·8-s + 8·10-s − 2·11-s + 4·13-s + 35·16-s − 2·17-s − 4·19-s + 20·20-s − 8·22-s − 8·23-s + 4·25-s + 16·26-s − 10·29-s − 8·31-s + 56·32-s − 8·34-s + 8·37-s − 16·38-s + 40·40-s − 6·41-s − 10·43-s − 20·44-s − 32·46-s − 12·47-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 0.894·5-s + 7.07·8-s + 2.52·10-s − 0.603·11-s + 1.10·13-s + 35/4·16-s − 0.485·17-s − 0.917·19-s + 4.47·20-s − 1.70·22-s − 1.66·23-s + 4/5·25-s + 3.13·26-s − 1.85·29-s − 1.43·31-s + 9.89·32-s − 1.37·34-s + 1.31·37-s − 2.59·38-s + 6.32·40-s − 0.937·41-s − 1.52·43-s − 3.01·44-s − 4.71·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.39675375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.39675375\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + 12 T^{3} - 29 T^{4} + 12 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 12 T^{2} - 12 T^{3} + 91 T^{4} - 12 p T^{5} - 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 7 T^{2} + 12 T^{3} + 152 T^{4} + 12 p T^{5} - 7 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 24 T^{2} - 12 T^{3} + 427 T^{4} - 12 p T^{5} - 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 30 T^{2} - 96 T^{3} - 845 T^{4} - 96 p T^{5} + 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 24 T^{2} + 180 T^{3} + 2035 T^{4} + 180 p T^{5} + 24 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} - 324 T^{3} - 2373 T^{4} - 324 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 22 T + 232 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 215 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 26 T + 304 T^{2} - 26 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 34 T^{2} - 4173 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 54 T^{2} - 576 T^{3} + 12667 T^{4} - 576 p T^{5} + 54 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 88 T^{2} + 324 T^{3} + 4251 T^{4} + 324 p T^{5} - 88 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 6 T - 55 T^{2} - 618 T^{3} - 4620 T^{4} - 618 p T^{5} - 55 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.82354877865124450663086070398, −6.72479872903577734837431487349, −6.49150907659239550835565813387, −6.27278171829400630598008707891, −6.09961200850969435708318059726, −5.74137361978059417326908259122, −5.67468550254103755322802732186, −5.24400367154001425098909853770, −5.16358319337739047583062230940, −5.08872631053590079798610718286, −5.01705651591322004394733772630, −4.52545422821642011329074667281, −4.14386015456911545012253653786, −3.71470549362742452654836125452, −3.69892932677045210678297604870, −3.69504557770311345886382816053, −3.66990128970961452086593701665, −3.06902715431995479093364851583, −2.61760043727950654055780612415, −2.23339136183512520727868701702, −2.07312648661058247143207900983, −2.04874632403757910557576302031, −1.89499811494096512355266646600, −1.08685075665687360093488358958, −0.61280032077115953288509947455,
0.61280032077115953288509947455, 1.08685075665687360093488358958, 1.89499811494096512355266646600, 2.04874632403757910557576302031, 2.07312648661058247143207900983, 2.23339136183512520727868701702, 2.61760043727950654055780612415, 3.06902715431995479093364851583, 3.66990128970961452086593701665, 3.69504557770311345886382816053, 3.69892932677045210678297604870, 3.71470549362742452654836125452, 4.14386015456911545012253653786, 4.52545422821642011329074667281, 5.01705651591322004394733772630, 5.08872631053590079798610718286, 5.16358319337739047583062230940, 5.24400367154001425098909853770, 5.67468550254103755322802732186, 5.74137361978059417326908259122, 6.09961200850969435708318059726, 6.27278171829400630598008707891, 6.49150907659239550835565813387, 6.72479872903577734837431487349, 6.82354877865124450663086070398
