L(s) = 1 | + 3·2-s + 4·4-s + 6·5-s + 3·8-s + 18·10-s − 2·13-s + 16-s + 24·20-s + 5·25-s − 6·26-s − 6·29-s + 6·32-s − 8·37-s + 18·40-s − 24·41-s − 19·49-s + 15·50-s − 8·52-s − 18·58-s − 2·61-s + 23·64-s − 12·65-s + 4·73-s − 24·74-s + 6·80-s − 72·82-s + 4·97-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 2·4-s + 2.68·5-s + 1.06·8-s + 5.69·10-s − 0.554·13-s + 1/4·16-s + 5.36·20-s + 25-s − 1.17·26-s − 1.11·29-s + 1.06·32-s − 1.31·37-s + 2.84·40-s − 3.74·41-s − 2.71·49-s + 2.12·50-s − 1.10·52-s − 2.36·58-s − 0.256·61-s + 23/8·64-s − 1.48·65-s + 0.468·73-s − 2.78·74-s + 0.670·80-s − 7.95·82-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.026190169\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.026190169\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + 5 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - 3 T + 11 T^{2} - 24 T^{3} + 54 T^{4} - 24 p T^{5} + 11 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 1558 p^{2} T^{10} + 181 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 7136 p^{2} T^{10} + 559 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + T - 17 T^{2} - 8 T^{3} + 142 T^{4} - 8 p T^{5} - 17 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 61 T^{2} + 1500 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 49 T^{2} + 1248 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 113498 p^{2} T^{10} + 3397 p^{4} T^{12} - 77 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 3 T + 59 T^{2} + 168 T^{3} + 2382 T^{4} + 168 p T^{5} + 59 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 13310 p^{2} T^{10} + 1345 p^{4} T^{12} + 55 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 12 T + 131 T^{2} + 996 T^{3} + 7176 T^{4} + 996 p T^{5} + 131 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 63424 p^{2} T^{10} - 593 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 194086 p^{2} T^{10} + 2053 p^{4} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 179704 p^{2} T^{10} - 617 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 8 p T^{5} - 113 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 1018720 p^{2} T^{10} + 10255 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 140 T^{2} + 10230 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - T + 138 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( 1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 379270 p^{2} T^{10} - 2555 p^{4} T^{12} + 115 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 2696642 p^{2} T^{10} + 22861 p^{4} T^{12} - 221 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 184 T^{2} + 21006 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 262 p T^{5} - 59 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−6.31981311775236219383435736590, −6.18673497785892180034177215634, −5.89979243042656340630716085674, −5.86643302778798293221600635166, −5.66996440415567541345914817885, −5.63949470376730856046578577406, −5.59094306053875691595832194730, −5.16577451337616603503425114637, −4.96917261267509067876460192179, −4.85673845474614830510142843406, −4.84435683226521457078002092218, −4.80397396391423472332909222377, −4.18782617368989494647911760662, −4.13811153091525444567144192665, −4.00679243161964543249884286083, −3.63474405634403271206662792801, −3.60477021559837359669690449450, −3.16107461221181382064433288203, −3.03796236864174000499559221724, −2.80018180209132222668056010980, −2.69303974174169208253702718443, −1.93384698792762223064612800871, −1.83737160902955030942261527507, −1.82256491519157782820972525716, −1.72379344189148610427166887432,
1.72379344189148610427166887432, 1.82256491519157782820972525716, 1.83737160902955030942261527507, 1.93384698792762223064612800871, 2.69303974174169208253702718443, 2.80018180209132222668056010980, 3.03796236864174000499559221724, 3.16107461221181382064433288203, 3.60477021559837359669690449450, 3.63474405634403271206662792801, 4.00679243161964543249884286083, 4.13811153091525444567144192665, 4.18782617368989494647911760662, 4.80397396391423472332909222377, 4.84435683226521457078002092218, 4.85673845474614830510142843406, 4.96917261267509067876460192179, 5.16577451337616603503425114637, 5.59094306053875691595832194730, 5.63949470376730856046578577406, 5.66996440415567541345914817885, 5.86643302778798293221600635166, 5.89979243042656340630716085674, 6.18673497785892180034177215634, 6.31981311775236219383435736590
Plot not available for L-functions of degree greater than 10.