| L(s) = 1 | − 4·5-s − 12·7-s + 6·9-s + 2·11-s + 8·13-s + 16·17-s + 12·19-s + 12·23-s + 11·25-s + 8·31-s + 48·35-s − 24·37-s − 18·41-s − 14·43-s − 24·45-s − 8·47-s + 71·49-s + 16·53-s − 8·55-s + 6·59-s − 12·61-s − 72·63-s − 32·65-s + 2·67-s − 24·77-s + 27·81-s + 22·83-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 4.53·7-s + 2·9-s + 0.603·11-s + 2.21·13-s + 3.88·17-s + 2.75·19-s + 2.50·23-s + 11/5·25-s + 1.43·31-s + 8.11·35-s − 3.94·37-s − 2.81·41-s − 2.13·43-s − 3.57·45-s − 1.16·47-s + 71/7·49-s + 2.19·53-s − 1.07·55-s + 0.781·59-s − 1.53·61-s − 9.07·63-s − 3.96·65-s + 0.244·67-s − 2.73·77-s + 3·81-s + 2.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653962141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653962141\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + p T^{2} - 8 T^{3} - 44 T^{4} - 8 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 300 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 26 T^{3} - 32 T^{4} - 26 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2976 T^{4} - 588 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 9 T^{2} - 156 T^{3} - 484 T^{4} - 156 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T + 13 T^{2} + 88 T^{3} - 344 T^{4} + 88 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 181 T^{2} + 1314 T^{3} + 8076 T^{4} + 1314 p T^{5} + 181 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 14 T + 65 T^{2} - 474 T^{3} - 6280 T^{4} - 474 p T^{5} + 65 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T - 19 T^{2} - 88 T^{3} + 1672 T^{4} - 88 p T^{5} - 19 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 1264 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 462 T^{3} + 848 T^{4} - 462 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 261 T^{2} + 2088 T^{3} + 24452 T^{4} + 2088 p T^{5} + 261 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 762 T^{3} + 2600 T^{4} - 762 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 22 T + 185 T^{2} + 290 T^{3} - 14024 T^{4} + 290 p T^{5} + 185 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.71037561023842606939950159584, −7.22278415579048469252497398337, −7.10152162371661289430432572657, −6.98380485137475481012919538917, −6.97870671830929623457396781753, −6.55039978667316210542565651752, −6.49352455696676240798388646126, −6.26951188858987718629264523414, −5.92065269526953227821907328908, −5.34878075618679775552245540164, −5.21670761898454019895349875639, −5.14998763732529670609669116467, −5.01640262029483827410255593502, −4.20893674299681912956562086341, −3.81038732303082324747259825074, −3.65851573183850815798317561239, −3.58397779739987362553225444999, −3.32459802285892343151666224042, −3.19185835578461986637065365405, −3.10205552391469698429463490078, −2.93578004228517419672645321092, −1.46631370867637442707384698035, −1.41062282073991015673508038785, −0.997629187816639011412711282623, −0.54060939883141623164365768143,
0.54060939883141623164365768143, 0.997629187816639011412711282623, 1.41062282073991015673508038785, 1.46631370867637442707384698035, 2.93578004228517419672645321092, 3.10205552391469698429463490078, 3.19185835578461986637065365405, 3.32459802285892343151666224042, 3.58397779739987362553225444999, 3.65851573183850815798317561239, 3.81038732303082324747259825074, 4.20893674299681912956562086341, 5.01640262029483827410255593502, 5.14998763732529670609669116467, 5.21670761898454019895349875639, 5.34878075618679775552245540164, 5.92065269526953227821907328908, 6.26951188858987718629264523414, 6.49352455696676240798388646126, 6.55039978667316210542565651752, 6.97870671830929623457396781753, 6.98380485137475481012919538917, 7.10152162371661289430432572657, 7.22278415579048469252497398337, 7.71037561023842606939950159584
