| L(s) = 1 | − 2·3-s − 6·4-s + 4·7-s + 4·9-s + 12·12-s + 13·16-s + 16·19-s − 8·21-s − 4·27-s − 24·28-s − 8·31-s − 24·36-s + 28·37-s + 36·43-s − 26·48-s + 8·49-s − 32·57-s + 28·61-s + 16·63-s − 6·64-s + 40·67-s + 28·73-s − 96·76-s + 16·79-s + 5·81-s + 48·84-s + 16·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3·4-s + 1.51·7-s + 4/3·9-s + 3.46·12-s + 13/4·16-s + 3.67·19-s − 1.74·21-s − 0.769·27-s − 4.53·28-s − 1.43·31-s − 4·36-s + 4.60·37-s + 5.48·43-s − 3.75·48-s + 8/7·49-s − 4.23·57-s + 3.58·61-s + 2.01·63-s − 3/4·64-s + 4.88·67-s + 3.27·73-s − 11.0·76-s + 1.80·79-s + 5/9·81-s + 5.23·84-s + 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.194208297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.194208297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 + 3 T^{2} + 7 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 22 T^{4} + 939 T^{8} - 22 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 2 T + 2 T^{2} + 24 T^{3} - 73 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 24 T^{2} + 338 T^{4} + 3504 T^{6} + 29907 T^{8} + 3504 p^{2} T^{10} + 338 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 38 T^{2} + 613 T^{4} - 9614 T^{6} + 189724 T^{8} - 9614 p^{2} T^{10} + 613 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 8 T + 20 T^{2} + 60 T^{3} - 649 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 + 34 T^{2} - 707 T^{4} + 6154 T^{6} + 1791292 T^{8} + 6154 p^{2} T^{10} - 707 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 4 T + 8 T^{2} + 36 T^{3} - 322 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 14 T + 113 T^{2} - 18 p T^{3} + 104 p T^{4} - 18 p^{2} T^{5} + 113 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 54 T^{2} + 221 T^{4} + 40554 T^{6} - 627828 T^{8} + 40554 p^{2} T^{10} + 221 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 18 T + 212 T^{2} - 1872 T^{3} + 13611 T^{4} - 1872 p T^{5} + 212 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 5500 T^{4} + 14557062 T^{8} - 5500 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 190 T^{2} + 14535 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 24 T^{2} - 3202 T^{4} - 81456 T^{6} + 70227 T^{8} - 81456 p^{2} T^{10} - 3202 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 20 T + 164 T^{2} - 564 T^{3} + 359 T^{4} - 564 p T^{5} + 164 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 24 T^{2} - 5554 T^{4} + 137904 T^{6} + 8572707 T^{8} + 137904 p^{2} T^{10} - 5554 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 14 T + 98 T^{2} - 1176 T^{3} + 13991 T^{4} - 1176 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 21212 T^{4} + 202731366 T^{8} + 21212 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 - 24 T^{2} + 9026 T^{4} - 212016 T^{6} + 16818147 T^{8} - 212016 p^{2} T^{10} + 9026 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 10 T + 2 p T^{2} + 2124 T^{3} + 23279 T^{4} + 2124 p T^{5} + 2 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.77898448597681675133552923683, −4.68028899608053317726174270188, −4.43171567307234226472407754072, −4.40809699269477077122043360799, −4.36308546912651289176207392987, −4.27807264469431567537230801161, −3.96024065086321508824707359009, −3.87090041505874895280465307412, −3.73030492021070528036463167990, −3.61061634248376991698046208201, −3.58302209362820257260827009076, −3.31833748058406673864514898220, −3.05290954055775619305096693903, −2.88517853406774644525367362005, −2.42736156810364704307919461499, −2.40366690777779871382732690868, −2.37419873775286964796565961732, −2.11921062660247027095508527948, −2.09096545732883971089114848175, −1.55517106438764427463189720281, −1.07502410790186060628515781433, −0.983634087168849504202564718852, −0.796253900134437259265293530987, −0.75728726497467023247403713233, −0.67182908395709811945723892526,
0.67182908395709811945723892526, 0.75728726497467023247403713233, 0.796253900134437259265293530987, 0.983634087168849504202564718852, 1.07502410790186060628515781433, 1.55517106438764427463189720281, 2.09096545732883971089114848175, 2.11921062660247027095508527948, 2.37419873775286964796565961732, 2.40366690777779871382732690868, 2.42736156810364704307919461499, 2.88517853406774644525367362005, 3.05290954055775619305096693903, 3.31833748058406673864514898220, 3.58302209362820257260827009076, 3.61061634248376991698046208201, 3.73030492021070528036463167990, 3.87090041505874895280465307412, 3.96024065086321508824707359009, 4.27807264469431567537230801161, 4.36308546912651289176207392987, 4.40809699269477077122043360799, 4.43171567307234226472407754072, 4.68028899608053317726174270188, 4.77898448597681675133552923683
Plot not available for L-functions of degree greater than 10.
