| L(s) = 1 | − 3·4-s + 24·9-s + 4·16-s − 8·29-s − 72·36-s + 28·49-s − 9·64-s + 324·81-s − 72·109-s + 24·116-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 96·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 8·9-s + 16-s − 1.48·29-s − 12·36-s + 4·49-s − 9/8·64-s + 36·81-s − 6.89·109-s + 2.22·116-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.607036273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.607036273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - p T^{2} )^{8} \) |
| 11 | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 16 T + 185 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 185 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | \( ( 1 + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 - p T^{2} )^{8} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 + p T^{2} )^{8} \) |
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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
−4.51013136184358079467491042034, −4.47121607501244428105723731900, −4.31375866704785903324585032319, −4.26030294860507599476217437847, −3.91017010440827035393522486591, −3.90407630548648826659765531099, −3.89914361361388208071361805573, −3.81034031616985835584705886749, −3.52605703607733055278328159876, −3.45541695835005750346648284906, −3.27074675249675852549957176073, −3.21844594314586152186467416367, −2.71257529886898354873588005543, −2.45423672349866192501160218437, −2.38769337068897655868670601316, −2.28790520958524141993235149578, −2.00169343661955720643783404275, −1.84174493706659170452486665831, −1.76950450970833662582545349352, −1.41430184714255724600427496352, −1.31879547352507982059918067938, −1.10706861593141603267919071731, −1.08741687263145026130161248014, −0.871912483507276550554667389136, −0.35718871683639559775363160860,
0.35718871683639559775363160860, 0.871912483507276550554667389136, 1.08741687263145026130161248014, 1.10706861593141603267919071731, 1.31879547352507982059918067938, 1.41430184714255724600427496352, 1.76950450970833662582545349352, 1.84174493706659170452486665831, 2.00169343661955720643783404275, 2.28790520958524141993235149578, 2.38769337068897655868670601316, 2.45423672349866192501160218437, 2.71257529886898354873588005543, 3.21844594314586152186467416367, 3.27074675249675852549957176073, 3.45541695835005750346648284906, 3.52605703607733055278328159876, 3.81034031616985835584705886749, 3.89914361361388208071361805573, 3.90407630548648826659765531099, 3.91017010440827035393522486591, 4.26030294860507599476217437847, 4.31375866704785903324585032319, 4.47121607501244428105723731900, 4.51013136184358079467491042034
Plot not available for L-functions of degree greater than 10.
