L(s) = 1 | − 4·2-s + 6·4-s + 12·5-s + 4·9-s − 48·10-s − 15·16-s − 16·18-s + 12·19-s + 72·20-s + 4·23-s + 74·25-s − 24·31-s + 24·32-s + 24·36-s − 48·38-s + 48·45-s − 16·46-s − 12·47-s − 12·49-s − 296·50-s − 20·53-s + 24·61-s + 96·62-s − 6·64-s + 72·76-s − 24·79-s − 180·80-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 3·4-s + 5.36·5-s + 4/3·9-s − 15.1·10-s − 3.75·16-s − 3.77·18-s + 2.75·19-s + 16.0·20-s + 0.834·23-s + 74/5·25-s − 4.31·31-s + 4.24·32-s + 4·36-s − 7.78·38-s + 7.15·45-s − 2.35·46-s − 1.75·47-s − 1.71·49-s − 41.8·50-s − 2.74·53-s + 3.07·61-s + 12.1·62-s − 3/4·64-s + 8.25·76-s − 2.70·79-s − 20.1·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.576644233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.576644233\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
good | 11 | \( 1 + 2 p T^{2} + 241 T^{4} + 2 p T^{6} - 14156 T^{8} + 2 p^{3} T^{10} + 241 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 6 T^{2} - 133 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 24 T^{2} + 287 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 828 T^{4} - 186 p T^{5} + 43 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 2 T - 13 T^{2} + 58 T^{3} - 332 T^{4} + 58 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 64 T^{2} + 2226 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 12 T + 112 T^{2} + 768 T^{3} + 4623 T^{4} + 768 p T^{5} + 112 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 90 T^{2} + 3817 T^{4} + 139050 T^{6} + 5381028 T^{8} + 139050 p^{2} T^{10} + 3817 p^{4} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 118 T^{2} + 6363 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 120 T^{2} + 6818 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 6 T + 19 T^{2} + 42 T^{3} - 1596 T^{4} + 42 p T^{5} + 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 100 T^{2} + 2458 T^{4} - 58000 T^{6} + 9954403 T^{8} - 58000 p^{2} T^{10} + 2458 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 36 T^{2} - 326 T^{4} - 264816 T^{6} - 23337981 T^{8} - 264816 p^{2} T^{10} - 326 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 232 T^{2} + 23058 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 136 T^{2} + 7534 T^{4} - 41344 T^{6} - 6797981 T^{8} - 41344 p^{2} T^{10} + 7534 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 12 T - 20 T^{2} + 72 T^{3} + 9279 T^{4} + 72 p T^{5} - 20 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 36 T^{2} - 3178 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 220 T^{2} + 22378 T^{4} - 2239600 T^{6} + 231025843 T^{8} - 2239600 p^{2} T^{10} + 22378 p^{4} T^{12} - 220 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 232 T^{2} + 27954 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−5.55894043339607881073954013777, −5.46474696832450576028279739796, −5.38577905637631234615851977469, −5.36529187534257736350517956506, −5.17934693072363739182501772432, −5.13143956615204619931651802965, −4.61914515318260965349340190319, −4.58347159173169057815828921323, −4.56475277346303233806449854941, −4.52090362550690704382364341755, −3.94612139210766564989398273095, −3.65663298282855415352594429622, −3.50378920485052429939015174518, −3.25481151172314200032636796302, −3.23733276215311334751537465814, −3.02337625024810887376488043207, −2.64887233708006081819641902319, −2.30666811375094055621543682092, −2.05832466972383533892055145329, −1.98595525714999236684361449843, −1.73942389558075121942672037980, −1.59737375030954305265428154860, −1.51991327009372258440640246719, −1.17597364674788496620234932764, −0.796904557158033602697725363193,
0.796904557158033602697725363193, 1.17597364674788496620234932764, 1.51991327009372258440640246719, 1.59737375030954305265428154860, 1.73942389558075121942672037980, 1.98595525714999236684361449843, 2.05832466972383533892055145329, 2.30666811375094055621543682092, 2.64887233708006081819641902319, 3.02337625024810887376488043207, 3.23733276215311334751537465814, 3.25481151172314200032636796302, 3.50378920485052429939015174518, 3.65663298282855415352594429622, 3.94612139210766564989398273095, 4.52090362550690704382364341755, 4.56475277346303233806449854941, 4.58347159173169057815828921323, 4.61914515318260965349340190319, 5.13143956615204619931651802965, 5.17934693072363739182501772432, 5.36529187534257736350517956506, 5.38577905637631234615851977469, 5.46474696832450576028279739796, 5.55894043339607881073954013777
Plot not available for L-functions of degree greater than 10.