| L(s) = 1 | + 4·2-s + 2·4-s − 20·8-s − 45·16-s + 16·32-s − 12·49-s + 16·53-s + 16·59-s + 204·64-s + 16·67-s − 16·83-s + 48·89-s − 48·98-s − 48·101-s + 32·103-s + 64·106-s + 64·118-s − 12·121-s + 127-s + 232·128-s + 131-s + 64·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4-s − 7.07·8-s − 11.2·16-s + 2.82·32-s − 1.71·49-s + 2.19·53-s + 2.08·59-s + 51/2·64-s + 1.95·67-s − 1.75·83-s + 5.08·89-s − 4.84·98-s − 4.77·101-s + 3.15·103-s + 6.21·106-s + 5.89·118-s − 1.09·121-s + 0.0887·127-s + 20.5·128-s + 0.0873·131-s + 5.52·134-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260115249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260115249\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 5 | $C_4\times C_2$ | \( 1 + 48 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 150 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 12 T^{2} - 474 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 96 T^{2} + 3888 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 108 T^{2} + 4806 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 48 T^{2} + 1392 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 144 T^{2} + 8448 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 104 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 48 T^{2} + 7920 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 204 T^{2} + 18918 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 192 T^{2} + 18624 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 108 T^{2} + 13830 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 288 T^{2} + 37632 T^{4} + 288 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.27812532141883533828071485175, −5.71677702565082683552872609190, −5.67532422556965809895655855280, −5.63729032742809265202420641321, −5.45962220202935630258034451776, −5.15858083191417923284527405552, −4.95588734833445889627201799211, −4.83006377224843928807455738385, −4.68859258686083234214962001559, −4.46961446154228375015318954889, −4.09523920694464165245193610704, −4.02046829821543776874038194076, −3.99092446381886370636163013730, −3.57954144728328508989846469615, −3.37043277890621611536788289633, −3.26631449237600780879302888059, −3.20301746708500458303548955190, −2.73621233909194990343543246469, −2.36609047756208253674464465735, −2.27028360766385487224714861411, −2.09786209232131405592120585234, −1.21772643226426224353855559022, −0.996301879110918194907453244003, −0.65429347814250614575934441933, −0.21218640636943875661005810440,
0.21218640636943875661005810440, 0.65429347814250614575934441933, 0.996301879110918194907453244003, 1.21772643226426224353855559022, 2.09786209232131405592120585234, 2.27028360766385487224714861411, 2.36609047756208253674464465735, 2.73621233909194990343543246469, 3.20301746708500458303548955190, 3.26631449237600780879302888059, 3.37043277890621611536788289633, 3.57954144728328508989846469615, 3.99092446381886370636163013730, 4.02046829821543776874038194076, 4.09523920694464165245193610704, 4.46961446154228375015318954889, 4.68859258686083234214962001559, 4.83006377224843928807455738385, 4.95588734833445889627201799211, 5.15858083191417923284527405552, 5.45962220202935630258034451776, 5.63729032742809265202420641321, 5.67532422556965809895655855280, 5.71677702565082683552872609190, 6.27812532141883533828071485175
