| L(s) = 1 | − 12·5-s + 10·7-s − 12·11-s + 48·17-s + 42·19-s + 24·23-s + 58·25-s − 102·31-s − 120·35-s + 22·37-s − 28·43-s − 132·47-s + 49·49-s − 120·53-s + 144·55-s − 24·59-s − 72·61-s − 110·67-s + 312·71-s − 66·73-s − 120·77-s + 10·79-s − 576·85-s + 72·89-s − 504·95-s − 36·101-s − 42·103-s + ⋯ |
| L(s) = 1 | − 2.39·5-s + 10/7·7-s − 1.09·11-s + 2.82·17-s + 2.21·19-s + 1.04·23-s + 2.31·25-s − 3.29·31-s − 3.42·35-s + 0.594·37-s − 0.651·43-s − 2.80·47-s + 49-s − 2.26·53-s + 2.61·55-s − 0.406·59-s − 1.18·61-s − 1.64·67-s + 4.39·71-s − 0.904·73-s − 1.55·77-s + 0.126·79-s − 6.77·85-s + 0.808·89-s − 5.30·95-s − 0.356·101-s − 0.407·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.579813818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.579813818\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T + 51 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 456 T^{3} + 2019 T^{4} + 456 p^{2} T^{5} + 86 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T - 85 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 48 T + 1514 T^{2} - 35808 T^{3} + 694947 T^{4} - 35808 p^{2} T^{5} + 1514 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 42 T + 1433 T^{2} - 35490 T^{3} + 795972 T^{4} - 35490 p^{2} T^{5} + 1433 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T + 22 T^{2} + 12096 T^{3} - 277629 T^{4} + 12096 p^{2} T^{5} + 22 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 102 T + 6041 T^{2} + 8466 p T^{3} + 9396 p^{2} T^{4} + 8466 p^{3} T^{5} + 6041 p^{4} T^{6} + 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 3674 p^{2} T^{5} - 2087 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 3675 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 132 T + 11654 T^{2} + 771672 T^{3} + 42125907 T^{4} + 771672 p^{2} T^{5} + 11654 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 120 T + 110 p T^{2} + 354240 T^{3} + 25104819 T^{4} + 354240 p^{2} T^{5} + 110 p^{5} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 6026 T^{2} + 140016 T^{3} + 22586547 T^{4} + 140016 p^{2} T^{5} + 6026 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 539280 p^{2} T^{5} + 9218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 110 T + 3625 T^{2} - 55330 T^{3} - 2642396 T^{4} - 55330 p^{2} T^{5} + 3625 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 156 T + 178 p T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 93786 p^{2} T^{5} + 2873 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 10 T - 3695 T^{2} + 86870 T^{3} - 25172156 T^{4} + 86870 p^{2} T^{5} - 3695 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 36 T + 8353 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.14353048933846558797273590033, −6.86748361962607562041309285228, −6.44295562451517239081153280380, −6.44070520269187062915379519426, −5.93388825130317141385494498370, −5.50414955676781437701264807342, −5.46956820473119886712766667810, −5.16574387279536563007796201393, −5.14135122716474241672923514819, −5.06742402666322240075023573700, −4.70220822036929088952846573615, −4.23320110082342686120197264991, −4.02857144301866381675283536811, −3.92214693263181214410570996600, −3.45425345903990108937269378947, −3.26171670497122543493099517573, −3.22897611241325566439599794010, −2.95920362816107700621719535298, −2.67756918131815117072728201818, −1.96350184430702811453529542267, −1.63899884315848400907841425132, −1.32825653985260738028535179054, −1.26998345302792874959749457181, −0.52302057431426142969610027722, −0.27462356762991584577016125107,
0.27462356762991584577016125107, 0.52302057431426142969610027722, 1.26998345302792874959749457181, 1.32825653985260738028535179054, 1.63899884315848400907841425132, 1.96350184430702811453529542267, 2.67756918131815117072728201818, 2.95920362816107700621719535298, 3.22897611241325566439599794010, 3.26171670497122543493099517573, 3.45425345903990108937269378947, 3.92214693263181214410570996600, 4.02857144301866381675283536811, 4.23320110082342686120197264991, 4.70220822036929088952846573615, 5.06742402666322240075023573700, 5.14135122716474241672923514819, 5.16574387279536563007796201393, 5.46956820473119886712766667810, 5.50414955676781437701264807342, 5.93388825130317141385494498370, 6.44070520269187062915379519426, 6.44295562451517239081153280380, 6.86748361962607562041309285228, 7.14353048933846558797273590033
