| L(s) = 1 | + 4-s − 2·5-s + 6·7-s + 9-s + 2·11-s + 4·13-s + 12·17-s + 6·19-s − 2·20-s + 18·23-s + 5·25-s + 6·28-s − 6·29-s + 16·31-s − 12·35-s + 36-s − 24·37-s + 8·41-s − 6·43-s + 2·44-s − 2·45-s + 8·49-s + 4·52-s − 4·55-s − 4·59-s − 4·61-s + 6·63-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 2.26·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 2.91·17-s + 1.37·19-s − 0.447·20-s + 3.75·23-s + 25-s + 1.13·28-s − 1.11·29-s + 2.87·31-s − 2.02·35-s + 1/6·36-s − 3.94·37-s + 1.24·41-s − 0.914·43-s + 0.301·44-s − 0.298·45-s + 8/7·49-s + 0.554·52-s − 0.539·55-s − 0.520·59-s − 0.512·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.730672263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.730672263\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 93 T^{2} - 540 T^{3} + 2552 T^{4} - 540 p T^{5} + 93 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 18 T + 180 T^{2} - 1296 T^{3} + 7139 T^{4} - 1296 p T^{5} + 180 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T - 19 T^{2} - 18 T^{3} + 1140 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 313 T^{2} + 2904 T^{3} + 20376 T^{4} + 2904 p T^{5} + 313 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 8 T - 7 T^{2} + 88 T^{3} + 736 T^{4} + 88 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 76 T^{2} + 384 T^{3} + 2763 T^{4} + 384 p T^{5} + 76 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 4442 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 9675 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T + 2 T^{2} - 416 T^{3} - 4229 T^{4} - 416 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 37 T^{2} - 572 T^{3} - 4256 T^{4} - 572 p T^{5} + 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 30 T + 508 T^{2} + 6240 T^{3} + 58875 T^{4} + 6240 p T^{5} + 508 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 112 T^{2} + 36 T^{3} + 14307 T^{4} + 36 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 8883 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 26186 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 4 T - 118 T^{2} + 176 T^{3} + 8611 T^{4} + 176 p T^{5} - 118 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.262733023524602418998763385469, −7.953973152663771466936379620920, −7.71532933453192850372235150110, −7.48172243868375387871380818157, −7.12715649430882503629909929381, −7.11702411408391041873356455339, −6.88671742328106557362757304172, −6.62908784082645664500683189537, −6.17433218108565313884331912840, −5.68557736043567521864881421311, −5.50040577500015953509313098496, −5.39932122297646622323811700157, −5.24236874838403678454331352579, −4.65192971937851753970201915713, −4.54732101826440316193153756058, −4.38926420482192687269236622296, −4.01808623356391780116952263342, −3.20279883338899669632961126412, −3.14543059982715082473191192345, −3.14375687815164612962176921181, −2.99382601072275777971945262796, −1.85688672543914951850933127166, −1.28366164638345805381266934476, −1.26775948270686756842543698650, −1.26486330681782281028356535555,
1.26486330681782281028356535555, 1.26775948270686756842543698650, 1.28366164638345805381266934476, 1.85688672543914951850933127166, 2.99382601072275777971945262796, 3.14375687815164612962176921181, 3.14543059982715082473191192345, 3.20279883338899669632961126412, 4.01808623356391780116952263342, 4.38926420482192687269236622296, 4.54732101826440316193153756058, 4.65192971937851753970201915713, 5.24236874838403678454331352579, 5.39932122297646622323811700157, 5.50040577500015953509313098496, 5.68557736043567521864881421311, 6.17433218108565313884331912840, 6.62908784082645664500683189537, 6.88671742328106557362757304172, 7.11702411408391041873356455339, 7.12715649430882503629909929381, 7.48172243868375387871380818157, 7.71532933453192850372235150110, 7.953973152663771466936379620920, 8.262733023524602418998763385469
