L(s) = 1 | + 2·2-s − 4·3-s + 4-s − 8·6-s − 4·7-s − 2·8-s + 6·9-s − 4·12-s − 4·13-s − 8·14-s − 4·16-s + 12·18-s + 20·19-s + 16·21-s + 8·24-s − 8·26-s + 4·27-s − 4·28-s − 4·31-s − 2·32-s + 6·36-s − 16·37-s + 40·38-s + 16·39-s − 18·41-s + 32·42-s − 10·43-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 2.30·3-s + 1/2·4-s − 3.26·6-s − 1.51·7-s − 0.707·8-s + 2·9-s − 1.15·12-s − 1.10·13-s − 2.13·14-s − 16-s + 2.82·18-s + 4.58·19-s + 3.49·21-s + 1.63·24-s − 1.56·26-s + 0.769·27-s − 0.755·28-s − 0.718·31-s − 0.353·32-s + 36-s − 2.63·37-s + 6.48·38-s + 2.56·39-s − 2.81·41-s + 4.93·42-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1529988986\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1529988986\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 8 T^{3} + 199 T^{4} - 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 8 p T^{5} - 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 190 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} - 144 T^{3} + 2259 T^{4} - 144 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 85 T^{2} - 18 T^{3} + 9036 T^{4} - 18 p T^{5} - 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 848 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 95 T^{2} + 190 T^{3} + 4 T^{4} + 190 p T^{5} - 95 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.994536546975601016031700817780, −7.53221918263658036481111378447, −7.27639666692337619615276747302, −7.11553883721982001271986254863, −6.85609045178531231853459727632, −6.72059832324328362776879037984, −6.49775552777731365659975152227, −6.24429939898981645973790688711, −5.77436337087557809150953141771, −5.53648767503993290039723138987, −5.51939764232879064184959526712, −5.30341507107101600564116017366, −5.03327263863342651316763184421, −4.84679228538997208371004382170, −4.82668155520275815234302295139, −4.12172909379042164114756744856, −3.85628211549987466967111300548, −3.26185493975143258739506416654, −3.25437106887929030614686478107, −3.22649891428358958167800965498, −2.85701557261642448488863660861, −2.13103791530114760922457750126, −1.49089697440882793314451398627, −1.01458189923360101643839017856, −0.15362727295134014143632646292,
0.15362727295134014143632646292, 1.01458189923360101643839017856, 1.49089697440882793314451398627, 2.13103791530114760922457750126, 2.85701557261642448488863660861, 3.22649891428358958167800965498, 3.25437106887929030614686478107, 3.26185493975143258739506416654, 3.85628211549987466967111300548, 4.12172909379042164114756744856, 4.82668155520275815234302295139, 4.84679228538997208371004382170, 5.03327263863342651316763184421, 5.30341507107101600564116017366, 5.51939764232879064184959526712, 5.53648767503993290039723138987, 5.77436337087557809150953141771, 6.24429939898981645973790688711, 6.49775552777731365659975152227, 6.72059832324328362776879037984, 6.85609045178531231853459727632, 7.11553883721982001271986254863, 7.27639666692337619615276747302, 7.53221918263658036481111378447, 7.994536546975601016031700817780