| L(s) = 1 | + 8·4-s − 2·9-s + 40·16-s + 8·19-s − 16·36-s − 56·49-s + 160·64-s + 64·76-s + 9·81-s − 28·121-s + 127-s + 131-s + 137-s + 139-s − 80·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s − 16·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4·4-s − 2/3·9-s + 10·16-s + 1.83·19-s − 8/3·36-s − 8·49-s + 20·64-s + 7.34·76-s + 81-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.66·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s − 1.22·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152556386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152556386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} )^{4} \) |
| 3 | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + p T^{2} )^{8} \) |
| 11 | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - p T^{2} )^{8} \) |
| 83 | \( ( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2}( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.72583909243822732444860140078, −4.59023768141134005300537684335, −4.43068662218473563290185324129, −4.25347043588027366066623516250, −3.95357851316239551897722462921, −3.64004634526643748522909168060, −3.61363964853908879114122741301, −3.51173267978459936367321768806, −3.47027351198616670300548116129, −3.40057104652322410010391382392, −3.30544441743702324893685626542, −2.91359136674415451820286099810, −2.82023621704821017627542107399, −2.74143437256550461782343291545, −2.55974332666153819736857269023, −2.45068201346103903184227581432, −2.28980101745073913005824646575, −2.12110011286142369096828815213, −1.77178395313481482222082709951, −1.70482468713934778770693159705, −1.38371864787430730591459735101, −1.29289675084234784794956700419, −1.18312129097478287061811640227, −1.10168728161086520050627443188, −0.091392521812361061427253712997,
0.091392521812361061427253712997, 1.10168728161086520050627443188, 1.18312129097478287061811640227, 1.29289675084234784794956700419, 1.38371864787430730591459735101, 1.70482468713934778770693159705, 1.77178395313481482222082709951, 2.12110011286142369096828815213, 2.28980101745073913005824646575, 2.45068201346103903184227581432, 2.55974332666153819736857269023, 2.74143437256550461782343291545, 2.82023621704821017627542107399, 2.91359136674415451820286099810, 3.30544441743702324893685626542, 3.40057104652322410010391382392, 3.47027351198616670300548116129, 3.51173267978459936367321768806, 3.61363964853908879114122741301, 3.64004634526643748522909168060, 3.95357851316239551897722462921, 4.25347043588027366066623516250, 4.43068662218473563290185324129, 4.59023768141134005300537684335, 4.72583909243822732444860140078
Plot not available for L-functions of degree greater than 10.
