| L(s) = 1 | − 4·3-s + 8·7-s + 8·9-s − 24·11-s + 16-s − 32·21-s − 8·27-s + 8·31-s + 96·33-s + 24·37-s + 12·41-s − 4·48-s + 32·49-s + 12·61-s + 64·63-s − 4·67-s + 16·73-s − 192·77-s − 7·81-s − 12·83-s − 32·93-s − 12·97-s − 192·99-s + 48·101-s − 8·103-s − 96·111-s + 8·112-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3.02·7-s + 8/3·9-s − 7.23·11-s + 1/4·16-s − 6.98·21-s − 1.53·27-s + 1.43·31-s + 16.7·33-s + 3.94·37-s + 1.87·41-s − 0.577·48-s + 32/7·49-s + 1.53·61-s + 8.06·63-s − 0.488·67-s + 1.87·73-s − 21.8·77-s − 7/9·81-s − 1.31·83-s − 3.31·93-s − 1.21·97-s − 19.2·99-s + 4.77·101-s − 0.788·103-s − 9.11·111-s + 0.755·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07609391400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07609391400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 8 T + 32 T^{2} - 64 T^{3} - p T^{4} + 464 T^{5} - 1440 T^{6} + 2472 T^{7} - 4016 T^{8} + 2472 p T^{9} - 1440 p^{2} T^{10} + 464 p^{3} T^{11} - p^{5} T^{12} - 64 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 12 T + 74 T^{2} + 312 T^{3} + 1083 T^{4} + 312 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 + 142 T^{4} - 8397 T^{8} + 142 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( 1 + 188 T^{4} - 45306 T^{8} + 188 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 32 T^{2} + 594 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 967 T^{4} + 655248 T^{8} - 967 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( 1 - 106 T^{2} + 6769 T^{4} - 295210 T^{6} + 9797332 T^{8} - 295210 p^{2} T^{10} + 6769 p^{4} T^{12} - 106 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 4 T - 44 T^{2} + 8 T^{3} + 2143 T^{4} + 8 p T^{5} - 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 6 T + 65 T^{2} - 318 T^{3} + 1620 T^{4} - 318 p T^{5} + 65 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 1778 T^{4} - 257517 T^{8} - 1778 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 4249 T^{4} + 13174320 T^{8} + 4249 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 - 4900 T^{4} + 13820838 T^{8} - 4900 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( 1 - 16 T^{2} - 5906 T^{4} + 12800 T^{6} + 24961747 T^{8} + 12800 p^{2} T^{10} - 5906 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 6 T - 89 T^{2} - 18 T^{3} + 9708 T^{4} - 18 p T^{5} - 89 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 4 T + 8 T^{2} - 304 T^{3} - 3511 T^{4} - 10624 T^{5} + 31800 T^{6} + 1654308 T^{7} - 4277600 T^{8} + 1654308 p T^{9} + 31800 p^{2} T^{10} - 10624 p^{3} T^{11} - 3511 p^{4} T^{12} - 304 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 244 T^{2} + 24582 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 32 T^{2} - 264 T^{3} + 578 T^{4} - 264 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 152 T^{2} + 16863 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 12 T + 72 T^{2} + 288 T^{3} - 8375 T^{4} - 56640 T^{5} - 35208 T^{6} + 3433884 T^{7} + 82789344 T^{8} + 3433884 p T^{9} - 35208 p^{2} T^{10} - 56640 p^{3} T^{11} - 8375 p^{4} T^{12} + 288 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 286 T^{2} + 35427 T^{4} + 286 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 12 T + 72 T^{2} - 744 T^{3} - 12542 T^{4} - 57996 T^{5} + 483840 T^{6} + 14345892 T^{7} + 128096019 T^{8} + 14345892 p T^{9} + 483840 p^{2} T^{10} - 57996 p^{3} T^{11} - 12542 p^{4} T^{12} - 744 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−4.88488864222204943626558000922, −4.88093741403543798096858458831, −4.80841345371285740254654329659, −4.64811039037155288436586154338, −4.56236686244949686314978786577, −4.44952686305174845534160114951, −4.27056287314976509738925932514, −3.88661342283345774116408205362, −3.87718060518937104688325882736, −3.52993927816232735562951592995, −3.50612487407875756643298282684, −3.05653247503336355870404156587, −3.03043074605402081035950539876, −2.58262951791814672215436990711, −2.56843301315613048337724061577, −2.52668089245270803036630115122, −2.45560667311653101304068440322, −2.38147510110409993279539022678, −2.08814307264416714205538695366, −1.85303847517883318616871046497, −1.37523153071072145077868778629, −1.13513997621708988247559467526, −0.822295225431753828815260983995, −0.78498707918688162994646362320, −0.081136362537314018819414080411,
0.081136362537314018819414080411, 0.78498707918688162994646362320, 0.822295225431753828815260983995, 1.13513997621708988247559467526, 1.37523153071072145077868778629, 1.85303847517883318616871046497, 2.08814307264416714205538695366, 2.38147510110409993279539022678, 2.45560667311653101304068440322, 2.52668089245270803036630115122, 2.56843301315613048337724061577, 2.58262951791814672215436990711, 3.03043074605402081035950539876, 3.05653247503336355870404156587, 3.50612487407875756643298282684, 3.52993927816232735562951592995, 3.87718060518937104688325882736, 3.88661342283345774116408205362, 4.27056287314976509738925932514, 4.44952686305174845534160114951, 4.56236686244949686314978786577, 4.64811039037155288436586154338, 4.80841345371285740254654329659, 4.88093741403543798096858458831, 4.88488864222204943626558000922
Plot not available for L-functions of degree greater than 10.
