| L(s) = 1 | − 2·7-s − 12·11-s + 4·23-s − 6·25-s − 4·37-s − 40·43-s + 2·49-s + 40·53-s + 40·67-s + 8·71-s + 24·77-s + 3·81-s − 32·107-s − 76·113-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 8·161-s + 163-s + 167-s + 173-s + 12·175-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 3.61·11-s + 0.834·23-s − 6/5·25-s − 0.657·37-s − 6.09·43-s + 2/7·49-s + 5.49·53-s + 4.88·67-s + 0.949·71-s + 2.73·77-s + 1/3·81-s − 3.09·107-s − 7.14·113-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.630·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.907·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3650227823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3650227823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 6 T^{2} + 18 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 + 2 T + 2 T^{2} - 26 T^{3} - 62 T^{4} - 26 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( 1 - p T^{4} - 92 T^{8} - p^{5} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 357 T^{4} + 68228 T^{8} + 357 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( 1 + 69 T^{4} - 53260 T^{8} + 69 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 18 T^{2} + 762 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 2 T + 2 T^{2} - 6 T^{3} - 382 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 83 T^{2} + 3148 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 18 T^{2} + 1962 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} + 34 T^{3} + 178 T^{4} + 34 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 106 T^{2} + 6130 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 - 2227 T^{4} + 6943924 T^{8} - 2227 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | \( ( 1 + 178 T^{2} + 14842 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T^{2} + 6822 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 14144 T^{4} + 103908670 T^{8} + 14144 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 271 T^{2} + 30340 T^{4} - 271 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 8400 T^{4} + 51732158 T^{8} - 8400 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 168 T^{2} + 14862 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 16837 T^{4} + 160234548 T^{8} + 16837 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−6.29960642786353366912424965233, −5.76321829058394977449345617334, −5.62853703861753949669252704998, −5.45183447578942555857929307049, −5.43164776908751106122162030743, −5.19811607378322860275989158549, −5.14458802819963715228890735626, −5.09351526967129225756535371441, −4.98231636437775924781913158767, −4.96258362900178377981288002298, −4.23585798116785910336953872767, −3.98015117411311510235667276860, −3.92713388405506045902296476051, −3.87686229710258682444746161906, −3.87169433268866987595556109588, −3.36113818588333654118962892379, −3.09141158211846384034313085558, −3.03438204165323222798640335287, −2.57525560097380973823581226295, −2.53491712664680416632808907380, −2.38766482791972815346043594420, −2.25113349903304211519715936473, −1.58709043505043042219040390675, −1.41588156006724541351922858563, −0.35259732372798149814835101820,
0.35259732372798149814835101820, 1.41588156006724541351922858563, 1.58709043505043042219040390675, 2.25113349903304211519715936473, 2.38766482791972815346043594420, 2.53491712664680416632808907380, 2.57525560097380973823581226295, 3.03438204165323222798640335287, 3.09141158211846384034313085558, 3.36113818588333654118962892379, 3.87169433268866987595556109588, 3.87686229710258682444746161906, 3.92713388405506045902296476051, 3.98015117411311510235667276860, 4.23585798116785910336953872767, 4.96258362900178377981288002298, 4.98231636437775924781913158767, 5.09351526967129225756535371441, 5.14458802819963715228890735626, 5.19811607378322860275989158549, 5.43164776908751106122162030743, 5.45183447578942555857929307049, 5.62853703861753949669252704998, 5.76321829058394977449345617334, 6.29960642786353366912424965233
Plot not available for L-functions of degree greater than 10.
