| L(s) = 1 | + 2·3-s − 4·5-s + 9-s + 4·11-s − 8·15-s + 4·17-s + 8·19-s + 4·23-s + 12·25-s − 2·27-s + 16·31-s + 8·33-s + 8·37-s − 8·41-s + 32·43-s − 4·45-s + 8·47-s + 8·51-s + 4·53-s − 16·55-s + 16·57-s + 16·59-s − 8·61-s − 16·67-s + 8·69-s − 8·71-s + 8·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1/3·9-s + 1.20·11-s − 2.06·15-s + 0.970·17-s + 1.83·19-s + 0.834·23-s + 12/5·25-s − 0.384·27-s + 2.87·31-s + 1.39·33-s + 1.31·37-s − 1.24·41-s + 4.87·43-s − 0.596·45-s + 1.16·47-s + 1.12·51-s + 0.549·53-s − 2.15·55-s + 2.11·57-s + 2.08·59-s − 1.02·61-s − 1.95·67-s + 0.963·69-s − 0.949·71-s + 0.936·73-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
| \(L(1)\) |
\(\approx\) |
\(9.047822304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.047822304\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 56 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T + 138 T^{2} - 896 T^{3} + 5027 T^{4} - 896 p T^{5} + 138 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_{4}$ | \( ( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T - 38 T^{2} - 64 T^{3} + 5187 T^{4} - 64 p T^{5} - 38 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 34 T^{2} + 496 T^{3} - 3333 T^{4} + 496 p T^{5} + 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} - 896 T^{3} + 11691 T^{4} - 896 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T + 24 T^{2} - 656 T^{3} - 6025 T^{4} - 656 p T^{5} + 24 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T - 48 T^{2} + 272 T^{3} + 2543 T^{4} + 272 p T^{5} - 48 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 4 T - 4 T^{2} - 632 T^{3} - 8945 T^{4} - 632 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.38465079230180221056826616440, −6.17864009440852319800875803842, −5.88985860945146900993556944987, −5.81995223188794844749664236469, −5.58711735384328896084579164136, −5.39489012244325391868188000894, −4.96156053222213495899242972913, −4.89278441993939403058011396713, −4.52059642562896008147224934009, −4.36646360424098482729497149772, −4.12719774092742370212296332237, −4.04006049474446831441921427451, −3.78193264035025920091634380083, −3.77573623908459863994608266061, −3.05971193608539679067963470247, −3.00778267007814502430851866402, −2.84504034134538881827147299755, −2.84306523921557145390910104862, −2.67263932808271957865276999013, −2.05681919804323685360089195547, −1.68351690274507679317685930558, −1.28865276233010613789576678004, −0.899765506938142028308698228459, −0.894114280058449976542923290038, −0.57090065081152910983353566370,
0.57090065081152910983353566370, 0.894114280058449976542923290038, 0.899765506938142028308698228459, 1.28865276233010613789576678004, 1.68351690274507679317685930558, 2.05681919804323685360089195547, 2.67263932808271957865276999013, 2.84306523921557145390910104862, 2.84504034134538881827147299755, 3.00778267007814502430851866402, 3.05971193608539679067963470247, 3.77573623908459863994608266061, 3.78193264035025920091634380083, 4.04006049474446831441921427451, 4.12719774092742370212296332237, 4.36646360424098482729497149772, 4.52059642562896008147224934009, 4.89278441993939403058011396713, 4.96156053222213495899242972913, 5.39489012244325391868188000894, 5.58711735384328896084579164136, 5.81995223188794844749664236469, 5.88985860945146900993556944987, 6.17864009440852319800875803842, 6.38465079230180221056826616440
