| L(s) = 1 | − 8·5-s − 8·13-s + 32·25-s − 8·29-s − 40·37-s + 24·49-s + 8·53-s − 40·61-s + 64·65-s + 64·97-s + 40·101-s + 40·109-s − 16·113-s − 88·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 3.57·5-s − 2.21·13-s + 32/5·25-s − 1.48·29-s − 6.57·37-s + 24/7·49-s + 1.09·53-s − 5.12·61-s + 7.93·65-s + 6.49·97-s + 3.98·101-s + 3.83·109-s − 1.50·113-s − 7.87·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6449595398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6449595398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 460 T^{4} + 82054 T^{8} - 460 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 4 T + 8 T^{2} + 44 T^{3} + 238 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 - 12 T^{4} - 247354 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 12 T^{2} + 1062 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 8 T^{2} - 20 T^{3} - 1106 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 11662 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 6068 T^{4} + 15827590 T^{8} + 6068 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T + 8 T^{2} - 76 T^{3} - 434 T^{4} - 76 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 12084 T^{4} + 60324614 T^{8} + 12084 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 20 T + 200 T^{2} + 1500 T^{3} + 11054 T^{4} + 1500 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 116 T^{4} - 21877946 T^{8} + 116 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 76 T^{2} + 9958 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 3468 T^{4} - 36032314 T^{8} - 3468 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 332 T^{2} + 43270 T^{4} - 332 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.44278370232028399581985600166, −3.32565087695571024815320955614, −3.25561439569282662553416812869, −3.21496414290937257495852961715, −3.15158620276726373268487186588, −2.86427956087731497967436309985, −2.66337967503140854416957333739, −2.60987173266799827374155599752, −2.53925616891296166537170518390, −2.50987456814370423180048380665, −2.10184230498567696993783739633, −2.03504050559872734682352834362, −2.03015138138485603592960724758, −1.97481990605412878085470046784, −1.90537814277342388703578281988, −1.61306924250978588142462380338, −1.41690830921585293162706525734, −1.28688801243571720454641933233, −1.15868430742329960546886703553, −0.981318543708285256240329210519, −0.71242652299642849485265853637, −0.45821407081926405887590847091, −0.38280853569707423071652775847, −0.26445304805312311599892229862, −0.18754767604481238141506645827,
0.18754767604481238141506645827, 0.26445304805312311599892229862, 0.38280853569707423071652775847, 0.45821407081926405887590847091, 0.71242652299642849485265853637, 0.981318543708285256240329210519, 1.15868430742329960546886703553, 1.28688801243571720454641933233, 1.41690830921585293162706525734, 1.61306924250978588142462380338, 1.90537814277342388703578281988, 1.97481990605412878085470046784, 2.03015138138485603592960724758, 2.03504050559872734682352834362, 2.10184230498567696993783739633, 2.50987456814370423180048380665, 2.53925616891296166537170518390, 2.60987173266799827374155599752, 2.66337967503140854416957333739, 2.86427956087731497967436309985, 3.15158620276726373268487186588, 3.21496414290937257495852961715, 3.25561439569282662553416812869, 3.32565087695571024815320955614, 3.44278370232028399581985600166
Plot not available for L-functions of degree greater than 10.
