| L(s) = 1 | + 2·2-s + 3·4-s + 2·8-s + 4·11-s + 2·13-s + 8·17-s + 8·22-s + 8·23-s + 2·25-s + 4·26-s − 4·29-s + 8·31-s − 6·32-s + 16·34-s − 8·37-s − 16·41-s − 8·43-s + 12·44-s + 16·46-s + 12·47-s + 6·49-s + 4·50-s + 6·52-s − 8·53-s − 8·58-s − 4·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.707·8-s + 1.20·11-s + 0.554·13-s + 1.94·17-s + 1.70·22-s + 1.66·23-s + 2/5·25-s + 0.784·26-s − 0.742·29-s + 1.43·31-s − 1.06·32-s + 2.74·34-s − 1.31·37-s − 2.49·41-s − 1.21·43-s + 1.80·44-s + 2.35·46-s + 1.75·47-s + 6/7·49-s + 0.565·50-s + 0.832·52-s − 1.09·53-s − 1.05·58-s − 0.520·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.71429524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.71429524\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 - 6 T^{2} - 13 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 64 p T^{5} - 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 16 T + 118 T^{2} + 896 T^{3} + 6867 T^{4} + 896 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} - 128 T^{3} + 299 T^{4} - 128 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 46 T^{2} - 48 T^{3} + 627 T^{4} - 48 p T^{5} + 46 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 74 T^{2} - 112 T^{3} + 3675 T^{4} - 112 p T^{5} - 74 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 18 T^{2} - 496 T^{3} - 4693 T^{4} - 496 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T - 78 T^{2} + 64 T^{3} + 11387 T^{4} + 64 p T^{5} - 78 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 30 T^{2} - 5341 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T - 122 T^{2} + 112 T^{3} + 10827 T^{4} + 112 p T^{5} - 122 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 150 T^{2} + 112 T^{3} + 16595 T^{4} + 112 p T^{5} - 150 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.85062812474068281997134737709, −6.84205373333418508643316693152, −6.80348239683497819347426909173, −6.37062448759988580936562266392, −6.12556907935562917298152013973, −6.11186227320858599313720963834, −5.46431723088367653490720018973, −5.36948367860488835685257758147, −5.33890087981600927099098883974, −5.33834253117311400815916410242, −4.69974980131245745962409037361, −4.53728428351395401629707022482, −4.40662984690695786390927191641, −4.03277231630527918776244654718, −3.58078653369610791702588306013, −3.55256939772973729453035609933, −3.20912441611431608167577365191, −3.11002453064044643207267840896, −3.08811002338321475648672428061, −2.44534883060535764075915599123, −2.00583185275968358663374787432, −1.66337252334658760666485527932, −1.63889493769614887654267736238, −0.807215398772624486932697466963, −0.77244955387151828656093101702,
0.77244955387151828656093101702, 0.807215398772624486932697466963, 1.63889493769614887654267736238, 1.66337252334658760666485527932, 2.00583185275968358663374787432, 2.44534883060535764075915599123, 3.08811002338321475648672428061, 3.11002453064044643207267840896, 3.20912441611431608167577365191, 3.55256939772973729453035609933, 3.58078653369610791702588306013, 4.03277231630527918776244654718, 4.40662984690695786390927191641, 4.53728428351395401629707022482, 4.69974980131245745962409037361, 5.33834253117311400815916410242, 5.33890087981600927099098883974, 5.36948367860488835685257758147, 5.46431723088367653490720018973, 6.11186227320858599313720963834, 6.12556907935562917298152013973, 6.37062448759988580936562266392, 6.80348239683497819347426909173, 6.84205373333418508643316693152, 6.85062812474068281997134737709
