| L(s) = 1 | − 8·3-s + 36·9-s − 120·27-s + 48·41-s − 16·43-s + 16·49-s + 48·67-s + 330·81-s − 16·83-s − 16·89-s + 32·107-s − 384·123-s + 127-s + 128·129-s + 131-s + 137-s + 139-s − 128·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 12·9-s − 23.0·27-s + 7.49·41-s − 2.43·43-s + 16/7·49-s + 5.86·67-s + 36.6·81-s − 1.75·83-s − 1.69·89-s + 3.09·107-s − 34.6·123-s + 0.0887·127-s + 11.2·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261658189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261658189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{8} \) |
| 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 - 8 T^{2} + 30 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 32 T^{2} + 510 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 24 T^{2} + 638 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - 48 T^{2} + 1502 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 80 T^{2} + 3582 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 144 T^{2} + 10046 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 176 T^{2} + 13950 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 212 T^{2} + 28710 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.52603258503569652376472699476, −4.22841036295965053707787847387, −4.10151636076806780420844223191, −3.99799179509554933589654530214, −3.83609255640133584770568582663, −3.76020687608706786517744825985, −3.67136262637149631619457737304, −3.58626220654420714593862702121, −3.50492594042962787816889581019, −3.01114756980248861315450659624, −2.77687933069760269909593992223, −2.68491446609497305019967750430, −2.57131597737954943385888648256, −2.56332600147851756376986605727, −2.26572542118183947364170604892, −2.17751550835024770514961136568, −1.73981160097029671869661774769, −1.69158491276635276746830240489, −1.48762224891828980202800188457, −1.42246539150618392954900810335, −0.842178854523292677140179373901, −0.792221961027751375156753631902, −0.77504470145550645523645446321, −0.61806894987921123948203027540, −0.32825710752831850795152923753,
0.32825710752831850795152923753, 0.61806894987921123948203027540, 0.77504470145550645523645446321, 0.792221961027751375156753631902, 0.842178854523292677140179373901, 1.42246539150618392954900810335, 1.48762224891828980202800188457, 1.69158491276635276746830240489, 1.73981160097029671869661774769, 2.17751550835024770514961136568, 2.26572542118183947364170604892, 2.56332600147851756376986605727, 2.57131597737954943385888648256, 2.68491446609497305019967750430, 2.77687933069760269909593992223, 3.01114756980248861315450659624, 3.50492594042962787816889581019, 3.58626220654420714593862702121, 3.67136262637149631619457737304, 3.76020687608706786517744825985, 3.83609255640133584770568582663, 3.99799179509554933589654530214, 4.10151636076806780420844223191, 4.22841036295965053707787847387, 4.52603258503569652376472699476
Plot not available for L-functions of degree greater than 10.
