| L(s) = 1 | − 17·16-s + 944·61-s − 162·81-s − 968·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 1.06·16-s + 15.4·61-s − 2·81-s − 8·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + 0.00418·239-s + 0.00414·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04188568972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04188568972\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 17 T^{4} + p^{8} T^{8} \) |
| 3 | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 13 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 21118 T^{4} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 550078 T^{4} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 31 | \( ( 1 - 2 T + p^{2} T^{2} )^{4}( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 37 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 43 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 8065922 T^{4} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 12619678 T^{4} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 61 | \( ( 1 - 118 T + p^{2} T^{2} )^{8} \) |
| 67 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 73 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 2878 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 3847202 T^{4} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 97 | \( ( 1 + p^{4} T^{4} )^{4} \) |
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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
−5.07703734072734632689554588900, −5.02357581666162163697548968083, −4.69582940329239365535334113111, −4.43747858285360837754523299048, −4.32098782739045583461543968297, −4.27935916635378175684146854658, −4.02716346567072023508153979067, −3.83775684583391264404675007900, −3.79069753304414447664902870563, −3.69877064546487989777035772545, −3.53993689354968975616312395183, −3.27634854878612855406112279831, −3.15922243004965806875682016909, −2.62807049693864140519501641514, −2.50540547546700592966998496545, −2.48488419577926880032402090163, −2.43643748932896669754995406389, −2.17293845804058851813231785752, −2.04502374222831571906696595921, −1.71599250869357659537540649365, −1.27968170410496016501764370162, −1.02091244536737276253716445506, −0.796058186908413338193523867679, −0.77795564418332026096952722812, −0.02289069366073654045357072169,
0.02289069366073654045357072169, 0.77795564418332026096952722812, 0.796058186908413338193523867679, 1.02091244536737276253716445506, 1.27968170410496016501764370162, 1.71599250869357659537540649365, 2.04502374222831571906696595921, 2.17293845804058851813231785752, 2.43643748932896669754995406389, 2.48488419577926880032402090163, 2.50540547546700592966998496545, 2.62807049693864140519501641514, 3.15922243004965806875682016909, 3.27634854878612855406112279831, 3.53993689354968975616312395183, 3.69877064546487989777035772545, 3.79069753304414447664902870563, 3.83775684583391264404675007900, 4.02716346567072023508153979067, 4.27935916635378175684146854658, 4.32098782739045583461543968297, 4.43747858285360837754523299048, 4.69582940329239365535334113111, 5.02357581666162163697548968083, 5.07703734072734632689554588900
Plot not available for L-functions of degree greater than 10.
