| L(s) = 1 | − 5-s + 3·7-s − 2·11-s − 5·13-s + 10·17-s − 14·19-s − 5·23-s + 2·25-s − 3·29-s + 7·31-s − 3·35-s + 12·37-s − 12·41-s + 8·43-s − 3·47-s + 8·49-s + 20·53-s + 2·55-s − 14·59-s + 61-s + 5·65-s + 4·67-s − 16·71-s − 14·73-s − 6·77-s − 7·79-s − 25·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 0.603·11-s − 1.38·13-s + 2.42·17-s − 3.21·19-s − 1.04·23-s + 2/5·25-s − 0.557·29-s + 1.25·31-s − 0.507·35-s + 1.97·37-s − 1.87·41-s + 1.21·43-s − 0.437·47-s + 8/7·49-s + 2.74·53-s + 0.269·55-s − 1.82·59-s + 0.128·61-s + 0.620·65-s + 0.488·67-s − 1.89·71-s − 1.63·73-s − 0.683·77-s − 0.787·79-s − 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063665755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063665755\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + T - T^{2} - 8 T^{3} - 26 T^{4} - 8 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3 T + T^{2} + 18 T^{3} - 48 T^{4} + 18 p T^{5} + p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 10 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 10 p T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 18 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 7 T - 17 T^{2} - 28 T^{3} + 1876 T^{4} - 28 p T^{5} - 17 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 59 T^{2} + 36 T^{3} - 360 T^{4} + 36 p T^{5} + 59 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T - 5 T^{2} + 136 T^{3} + 160 T^{4} + 136 p T^{5} - 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3 T - 79 T^{2} - 18 T^{3} + 5112 T^{4} - 18 p T^{5} - 79 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 8 p T^{5} - 113 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 89 T^{2} + 116 T^{3} + 5464 T^{4} + 116 p T^{5} - 89 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 113 T^{2} + 28 T^{3} + 16132 T^{4} + 28 p T^{5} - 113 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 25 T + 311 T^{2} + 3700 T^{3} + 39832 T^{4} + 3700 p T^{5} + 311 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 136 p T^{5} - 113 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.219677076636985472743014439762, −7.79693966259413327952596571353, −7.58840994093164300035617255103, −7.38534206946627289358424948808, −7.23206857939337731665914599841, −6.97231378291655631167487961381, −6.67351564396517354471419214895, −6.13013389195367265341969304168, −6.04022117045450026440894952580, −5.86635779328315002377185678761, −5.48880192083148248739817997614, −5.40927218254777713457933135498, −4.91089788946891995742209696706, −4.55430624835628224038225231066, −4.45555077688467848645999613343, −4.25466852886473431711525844321, −4.00574712927983715587259694626, −3.58976871816784399937698576073, −3.05397100992546336623883536421, −2.79205708030740069808324107060, −2.41053796528627183459936871492, −2.31146940746556216686206464412, −1.47666566866945140604749178685, −1.44560055731205528099225838787, −0.37374736720858796170290176430,
0.37374736720858796170290176430, 1.44560055731205528099225838787, 1.47666566866945140604749178685, 2.31146940746556216686206464412, 2.41053796528627183459936871492, 2.79205708030740069808324107060, 3.05397100992546336623883536421, 3.58976871816784399937698576073, 4.00574712927983715587259694626, 4.25466852886473431711525844321, 4.45555077688467848645999613343, 4.55430624835628224038225231066, 4.91089788946891995742209696706, 5.40927218254777713457933135498, 5.48880192083148248739817997614, 5.86635779328315002377185678761, 6.04022117045450026440894952580, 6.13013389195367265341969304168, 6.67351564396517354471419214895, 6.97231378291655631167487961381, 7.23206857939337731665914599841, 7.38534206946627289358424948808, 7.58840994093164300035617255103, 7.79693966259413327952596571353, 8.219677076636985472743014439762
