| L(s) = 1 | − 164·4-s + 1.04e3·9-s + 5.44e4·13-s + 1.07e4·16-s + 2.14e5·25-s − 1.71e5·36-s − 2.17e4·37-s + 3.77e6·49-s − 8.92e6·52-s − 3.02e6·61-s − 1.80e6·64-s + 3.68e6·73-s + 4.70e6·81-s − 1.58e7·97-s − 3.52e7·100-s − 6.71e7·109-s + 5.68e7·117-s − 3.01e7·121-s + 127-s + 131-s + 137-s + 139-s + 1.12e7·144-s + 3.57e6·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.28·4-s + 0.477·9-s + 6.87·13-s + 0.657·16-s + 2.74·25-s − 0.611·36-s − 0.0706·37-s + 4.58·49-s − 8.80·52-s − 1.70·61-s − 0.862·64-s + 1.10·73-s + 0.984·81-s − 1.76·97-s − 3.52·100-s − 4.96·109-s + 3.28·117-s − 1.54·121-s + 0.313·144-s + 0.0905·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.348391240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.348391240\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 41 p^{2} T^{2} + 63 p^{8} T^{4} + 41 p^{16} T^{6} + p^{28} T^{8} \) |
| 3 | \( 1 - 116 p^{2} T^{2} - 14894 p^{5} T^{4} - 116 p^{16} T^{6} + p^{28} T^{8} \) |
| good | 5 | \( ( 1 - 21476 p T^{2} + 70176174 p^{3} T^{4} - 21476 p^{15} T^{6} + p^{28} T^{8} )^{2} \) |
| 7 | \( ( 1 - 1886596 T^{2} + 2237391652518 T^{4} - 1886596 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 11 | \( ( 1 + 15051980 T^{2} + 46104248605878 T^{4} + 15051980 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 13 | \( ( 1 - 13612 T + 13101798 p T^{2} - 13612 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 17 | \( ( 1 - 1564780996 T^{2} + 948082382592036678 T^{4} - 1564780996 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2363096020 T^{2} + 2897086659724142358 T^{4} - 2363096020 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 23 | \( ( 1 + 10655298716 T^{2} + 50911417411485435558 T^{4} + 10655298716 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 29 | \( ( 1 - 15119114260 T^{2} + \)\(42\!\cdots\!38\)\( T^{4} - 15119114260 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 31 | \( ( 1 - 92139890404 T^{2} + \)\(35\!\cdots\!46\)\( T^{4} - 92139890404 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 37 | \( ( 1 + 5444 T + 86443034526 T^{2} + 5444 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 41 | \( ( 1 - 670723359844 T^{2} + \)\(18\!\cdots\!06\)\( T^{4} - 670723359844 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 43 | \( ( 1 - 281458444084 T^{2} + \)\(11\!\cdots\!18\)\( T^{4} - 281458444084 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 47 | \( ( 1 + 1594540776764 T^{2} + \)\(11\!\cdots\!18\)\( T^{4} + 1594540776764 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 53 | \( ( 1 - 3369637018804 T^{2} + \)\(51\!\cdots\!38\)\( T^{4} - 3369637018804 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 59 | \( ( 1 + 5549519857676 T^{2} + \)\(17\!\cdots\!66\)\( T^{4} + 5549519857676 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 61 | \( ( 1 + 756980 T + 3358166464398 T^{2} + 756980 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 67 | \( ( 1 - 1556381570836 T^{2} + \)\(48\!\cdots\!98\)\( T^{4} - 1556381570836 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 71 | \( ( 1 + 11818476110300 T^{2} + \)\(50\!\cdots\!38\)\( T^{4} + 11818476110300 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 73 | \( ( 1 - 920692 T + 21663235778454 T^{2} - 920692 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 79 | \( ( 1 - 27183610132900 T^{2} + \)\(70\!\cdots\!78\)\( T^{4} - 27183610132900 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 83 | \( ( 1 + 92589213551276 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} + 92589213551276 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 89 | \( ( 1 - 152646663101860 T^{2} + \)\(96\!\cdots\!78\)\( T^{4} - 152646663101860 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 97 | \( ( 1 + 3971804 T + 152677108318086 T^{2} + 3971804 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
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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
−8.620257697020202861689405376855, −8.489980765881844648413441421905, −8.350783025181844883130460889016, −7.958837347027837566877257688645, −7.899245616175774624422635597781, −7.25164007334784795019689063093, −7.06290325708507056494841362074, −6.73010153761778223638224265107, −6.46083667002456922951709334132, −6.13458049156249601218468601606, −5.99387887763525521322903254639, −5.89611415433032889647094406103, −5.28246757714233321359983068434, −5.19781598998420420516834956457, −4.66845148338536239992438860703, −4.06809259066615237080316720318, −3.98053879207646576144603953310, −3.82154877893615284321762027149, −3.58888682966595410436067759997, −3.01022716062072110820166711668, −2.56911216979886762273278787741, −1.30979746638786411429462279956, −1.27298276528861622387522654202, −1.17235647672031556261493035009, −0.57268461090912883470649677242,
0.57268461090912883470649677242, 1.17235647672031556261493035009, 1.27298276528861622387522654202, 1.30979746638786411429462279956, 2.56911216979886762273278787741, 3.01022716062072110820166711668, 3.58888682966595410436067759997, 3.82154877893615284321762027149, 3.98053879207646576144603953310, 4.06809259066615237080316720318, 4.66845148338536239992438860703, 5.19781598998420420516834956457, 5.28246757714233321359983068434, 5.89611415433032889647094406103, 5.99387887763525521322903254639, 6.13458049156249601218468601606, 6.46083667002456922951709334132, 6.73010153761778223638224265107, 7.06290325708507056494841362074, 7.25164007334784795019689063093, 7.899245616175774624422635597781, 7.958837347027837566877257688645, 8.350783025181844883130460889016, 8.489980765881844648413441421905, 8.620257697020202861689405376855
Plot not available for L-functions of degree greater than 10.
