| L(s) = 1 | + 2-s + 3·4-s + 4·7-s + 4·8-s + 9-s − 4·11-s + 4·13-s + 4·14-s + 6·16-s + 2·17-s + 18-s − 4·19-s − 4·22-s + 12·23-s + 4·26-s + 12·28-s + 6·29-s + 6·32-s + 2·34-s + 3·36-s + 6·37-s − 4·38-s − 6·41-s − 8·43-s − 12·44-s + 12·46-s − 16·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 3/2·4-s + 1.51·7-s + 1.41·8-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 1.06·14-s + 3/2·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.852·22-s + 2.50·23-s + 0.784·26-s + 2.26·28-s + 1.11·29-s + 1.06·32-s + 0.342·34-s + 1/2·36-s + 0.986·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s − 1.80·44-s + 1.76·46-s − 2.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.040552917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.040552917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T - p T^{2} + T^{3} + 3 T^{4} + p T^{5} - p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 4 T^{3} + 8 T^{4} + 4 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 5 T^{2} - 4 T^{3} + 144 T^{4} - 4 p T^{5} - 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 11 T^{2} + 38 T^{3} - 132 T^{4} + 38 p T^{5} - 11 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 21 T^{2} - 4 T^{3} + 704 T^{4} - 4 p T^{5} - 21 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 11 T^{2} + 66 T^{3} + 324 T^{4} + 66 p T^{5} - 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T + 25 T^{2} - 426 T^{3} - 3036 T^{4} - 426 p T^{5} + 25 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 33 T^{2} + 88 T^{3} + 4808 T^{4} + 88 p T^{5} - 33 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 35 T^{2} - 108 T^{3} - 96 T^{4} - 108 p T^{5} + 35 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 41 T^{2} + 232 T^{3} + 2248 T^{4} + 232 p T^{5} - 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8 T - 89 T^{2} - 88 T^{3} + 13824 T^{4} - 88 p T^{5} - 89 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 171 T^{2} + 38 T^{3} + 20828 T^{4} + 38 p T^{5} - 171 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.040107285992264161983795794839, −8.029868090796296025913096924731, −7.972661243778127687152663135384, −7.946667428119654024854326274085, −7.55347861868168264227253097549, −6.94088818195919130485061650653, −6.73023633841948868098698111339, −6.66629651106512238511901918883, −6.53467463696251984531624811190, −6.28172856525808172723090059354, −5.71608149209922709986782640188, −5.33628108275743659552243867004, −5.18610495781063016831296549462, −5.12048435712763670274353868291, −4.69258707770257581007072838637, −4.47698604952670919961962973509, −4.22859764591045912876984487004, −3.60236814137968309890456470238, −3.21905557787477084814630647978, −3.17444443030196359609939706780, −2.77478689054182247493250064317, −2.05139155205742518911888111467, −2.03214319651505842861828524135, −1.41914531227337058377371048762, −1.07479646069549256135632957733,
1.07479646069549256135632957733, 1.41914531227337058377371048762, 2.03214319651505842861828524135, 2.05139155205742518911888111467, 2.77478689054182247493250064317, 3.17444443030196359609939706780, 3.21905557787477084814630647978, 3.60236814137968309890456470238, 4.22859764591045912876984487004, 4.47698604952670919961962973509, 4.69258707770257581007072838637, 5.12048435712763670274353868291, 5.18610495781063016831296549462, 5.33628108275743659552243867004, 5.71608149209922709986782640188, 6.28172856525808172723090059354, 6.53467463696251984531624811190, 6.66629651106512238511901918883, 6.73023633841948868098698111339, 6.94088818195919130485061650653, 7.55347861868168264227253097549, 7.946667428119654024854326274085, 7.972661243778127687152663135384, 8.029868090796296025913096924731, 8.040107285992264161983795794839
