| L(s) = 1 | − 2·3-s − 4·5-s + 9-s − 4·11-s + 8·15-s − 12·17-s + 8·19-s − 4·23-s + 12·25-s + 2·27-s + 8·33-s − 8·37-s + 24·41-s − 4·45-s + 8·47-s + 24·51-s + 4·53-s + 16·55-s − 16·57-s − 8·61-s + 8·69-s + 8·71-s − 24·73-s − 24·75-s + 16·79-s − 4·81-s + 16·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s + 2.06·15-s − 2.91·17-s + 1.83·19-s − 0.834·23-s + 12/5·25-s + 0.384·27-s + 1.39·33-s − 1.31·37-s + 3.74·41-s − 0.596·45-s + 1.16·47-s + 3.36·51-s + 0.549·53-s + 2.15·55-s − 2.11·57-s − 1.02·61-s + 0.963·69-s + 0.949·71-s − 2.80·73-s − 2.77·75-s + 1.80·79-s − 4/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6157344251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6157344251\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 76 T^{2} + 24 p T^{3} + 111 p T^{4} + 24 p^{2} T^{5} + 76 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} - 64 T^{3} + 539 T^{4} - 64 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 54 T^{2} + 1955 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T + 26 T^{2} + 448 T^{3} - 3773 T^{4} + 448 p T^{5} + 26 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 46 T^{2} - 1365 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T - 24 T^{2} - 272 T^{3} + 119 T^{4} - 272 p T^{5} - 24 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 6 T^{2} - 4453 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 24 T + 304 T^{2} + 3024 T^{3} + 26607 T^{4} + 3024 p T^{5} + 304 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 66 T^{2} - 512 T^{3} + 9635 T^{4} - 512 p T^{5} + 66 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 1640 p T^{5} + 140 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.42857366738149936235763965929, −6.08462351024484500938163650188, −5.96652619489415415363002759330, −5.74539201605017353001193258709, −5.52377100147487369750127577983, −5.38838712614704163833718543131, −4.93744783841756653642825520495, −4.92410591286054567394627194626, −4.87671875123341616288257802917, −4.39357527582527998993836045578, −4.25416431308752514173811881619, −4.13534364043385130907435545471, −3.90795473054530167592804226747, −3.62958049169295949889370623614, −3.49929254197255194792370720493, −2.90494128923396826989141726653, −2.79875017411983148017027612105, −2.63005227129464675761450515195, −2.49014090715462192335177686223, −2.11026284953365007849208036559, −1.53170232092274488988036953495, −1.45752086628693038239251068919, −0.74614105882130844943370644350, −0.55722235755361820308712145489, −0.27393437285472496430538448632,
0.27393437285472496430538448632, 0.55722235755361820308712145489, 0.74614105882130844943370644350, 1.45752086628693038239251068919, 1.53170232092274488988036953495, 2.11026284953365007849208036559, 2.49014090715462192335177686223, 2.63005227129464675761450515195, 2.79875017411983148017027612105, 2.90494128923396826989141726653, 3.49929254197255194792370720493, 3.62958049169295949889370623614, 3.90795473054530167592804226747, 4.13534364043385130907435545471, 4.25416431308752514173811881619, 4.39357527582527998993836045578, 4.87671875123341616288257802917, 4.92410591286054567394627194626, 4.93744783841756653642825520495, 5.38838712614704163833718543131, 5.52377100147487369750127577983, 5.74539201605017353001193258709, 5.96652619489415415363002759330, 6.08462351024484500938163650188, 6.42857366738149936235763965929
