| L(s) = 1 | + 4·7-s + 2·9-s − 20·25-s − 16·37-s + 32·43-s + 16·49-s + 8·63-s − 16·67-s − 56·79-s + 2·81-s − 32·109-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 76·169-s + 173-s − 80·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2/3·9-s − 4·25-s − 2.63·37-s + 4.87·43-s + 16/7·49-s + 1.00·63-s − 1.95·67-s − 6.30·79-s + 2/9·81-s − 3.06·109-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.84·169-s + 0.0760·173-s − 6.04·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.125493053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125493053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T^{2} + 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 2 p T^{2} + 58 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 38 T^{2} + 682 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 48 T^{2} + 1086 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T^{2} + 706 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 32 T^{2} + 238 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 32 T^{2} + 2110 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 48 T^{2} + 606 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 4 T^{2} + 4150 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 110 T^{2} + 8610 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 118 T^{2} + 9546 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 88 T^{2} + 6510 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 132 T^{2} + 10662 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 318 T^{2} + 39042 T^{4} + 318 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 24 T^{2} + 12654 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.26783455915485210584904547105, −4.93099474240491624038286019965, −4.78794072107174549931638044059, −4.70210203910250901032873659074, −4.56138455467781866978015450484, −4.32381133816293553943248218945, −4.18698246745623218711546618449, −4.05486148682557574749313775436, −3.99105783358831616684789652388, −3.96342024585865910414097724900, −3.92667060048296839018357880969, −3.46968088782672426703142006458, −3.18333998518573772550815199773, −3.11629565878178469884977726177, −2.89131311290160942616358338992, −2.79667674273152209371847369246, −2.48010243869065890304008324598, −2.26899033263381221078384950345, −2.12691233591402441622421286357, −1.79930769872408241502212414992, −1.73904844512396491850889186250, −1.44108730037522169877312617997, −1.42461815282329129469324100786, −0.854972866519842845132605114597, −0.33789138564938135690419200069,
0.33789138564938135690419200069, 0.854972866519842845132605114597, 1.42461815282329129469324100786, 1.44108730037522169877312617997, 1.73904844512396491850889186250, 1.79930769872408241502212414992, 2.12691233591402441622421286357, 2.26899033263381221078384950345, 2.48010243869065890304008324598, 2.79667674273152209371847369246, 2.89131311290160942616358338992, 3.11629565878178469884977726177, 3.18333998518573772550815199773, 3.46968088782672426703142006458, 3.92667060048296839018357880969, 3.96342024585865910414097724900, 3.99105783358831616684789652388, 4.05486148682557574749313775436, 4.18698246745623218711546618449, 4.32381133816293553943248218945, 4.56138455467781866978015450484, 4.70210203910250901032873659074, 4.78794072107174549931638044059, 4.93099474240491624038286019965, 5.26783455915485210584904547105
Plot not available for L-functions of degree greater than 10.
