| L(s) = 1 | + 71·9-s − 1.07e3·19-s − 1.02e3·31-s + 1.36e4·49-s + 2.81e3·61-s + 3.58e4·79-s − 6.33e3·81-s + 2.93e4·109-s + 8.91e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.15e5·169-s − 7.61e4·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 0.876·9-s − 2.96·19-s − 1.06·31-s + 5.68·49-s + 0.755·61-s + 5.74·79-s − 0.965·81-s + 2.47·109-s + 6.08·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 7.53·169-s − 2.60·171-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(31.50452277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.50452277\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 71 T^{2} + 1264 p^{2} T^{4} - 71 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 6827 T^{2} + 21855828 T^{4} - 6827 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 11 | \( ( 1 - 44569 T^{2} + 905592696 T^{4} - 44569 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 13 | \( ( 1 - 107567 T^{2} + 4514359008 T^{4} - 107567 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 + 3137 p T^{2} + 8506428936 T^{4} + 3137 p^{9} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 + 268 T + 8967 p T^{2} + 268 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 23 | \( ( 1 + 638944 T^{2} + 210680481246 T^{4} + 638944 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2033344 T^{2} + 1876708445406 T^{4} - 2033344 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 31 | \( ( 1 + 257 T + 1187148 T^{2} + 257 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 37 | \( ( 1 - 6675296 T^{2} + 18080662254846 T^{4} - 6675296 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 + 2803511 T^{2} + 8571368969016 T^{4} + 2803511 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 43 | \( ( 1 - 9244247 T^{2} + 39988212098448 T^{4} - 9244247 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 + 2130904 T^{2} - 22645911464274 T^{4} + 2130904 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 + 7515904 T^{2} - 2039110327074 T^{4} + 7515904 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 21375124 T^{2} + 333957387782886 T^{4} - 21375124 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 61 | \( ( 1 - 703 T - 1649022 T^{2} - 703 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 67 | \( ( 1 - 60536402 T^{2} + 1725904585648083 T^{4} - 60536402 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4594496 T^{2} - 176968404991074 T^{4} + 4594496 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 - 18096791 T^{2} + 1439785542534576 T^{4} - 18096791 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8966 T + 89231226 T^{2} - 8966 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 83 | \( ( 1 + 34367329 T^{2} - 1041395369052864 T^{4} + 34367329 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 89 | \( ( 1 - 130574209 T^{2} + 8619795270816576 T^{4} - 130574209 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 - 241246607 T^{2} + 27040018542241728 T^{4} - 241246607 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
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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
−4.34014297491875959509400786888, −4.27030830615244858879065483125, −4.19821096929189704171662476434, −4.17866556598601899544168283423, −4.14708507499319800809248570429, −3.99366428474546253942523395680, −3.53648308839261883352861686831, −3.41634987808896144944930192195, −3.12140979881440703739702835893, −3.10446182723556705530026189054, −3.01175085892782193736119857455, −2.88687943561224766746329635085, −2.52393433965323572185591890591, −2.16561377060972695643403639839, −1.95025930683675206149862631134, −1.91393744114653557788357292199, −1.89044525955583822929667288477, −1.86652306707827897966763587666, −1.86549675644930593551233844390, −0.994247599960904333744778054978, −0.811369507753793039690614445265, −0.62519962872044008455338457987, −0.61457524012478712960367692796, −0.57655271093456731942240186135, −0.39178523930878606851914966843,
0.39178523930878606851914966843, 0.57655271093456731942240186135, 0.61457524012478712960367692796, 0.62519962872044008455338457987, 0.811369507753793039690614445265, 0.994247599960904333744778054978, 1.86549675644930593551233844390, 1.86652306707827897966763587666, 1.89044525955583822929667288477, 1.91393744114653557788357292199, 1.95025930683675206149862631134, 2.16561377060972695643403639839, 2.52393433965323572185591890591, 2.88687943561224766746329635085, 3.01175085892782193736119857455, 3.10446182723556705530026189054, 3.12140979881440703739702835893, 3.41634987808896144944930192195, 3.53648308839261883352861686831, 3.99366428474546253942523395680, 4.14708507499319800809248570429, 4.17866556598601899544168283423, 4.19821096929189704171662476434, 4.27030830615244858879065483125, 4.34014297491875959509400786888
Plot not available for L-functions of degree greater than 10.
