| L(s) = 1 | − 3·5-s − 3·11-s − 3·13-s − 6·17-s + 3·19-s − 3·23-s + 6·25-s − 6·27-s + 12·29-s + 12·31-s + 9·37-s − 9·41-s − 12·43-s − 15·47-s + 9·53-s + 9·55-s + 24·59-s + 6·61-s + 9·65-s − 6·67-s + 18·79-s − 30·83-s + 18·85-s − 9·95-s + 6·97-s + 18·101-s − 6·103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.904·11-s − 0.832·13-s − 1.45·17-s + 0.688·19-s − 0.625·23-s + 6/5·25-s − 1.15·27-s + 2.22·29-s + 2.15·31-s + 1.47·37-s − 1.40·41-s − 1.82·43-s − 2.18·47-s + 1.23·53-s + 1.21·55-s + 3.12·59-s + 0.768·61-s + 1.11·65-s − 0.733·67-s + 2.02·79-s − 3.29·83-s + 1.95·85-s − 0.923·95-s + 0.609·97-s + 1.79·101-s − 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.051780179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.051780179\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{6}$ | \( 1 + 2 p T^{3} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 9 T^{2} + 2 p T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 10 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 130 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 54 T^{2} + 145 T^{3} + 54 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 108 T^{2} - 670 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 12 T + 105 T^{2} - 616 T^{3} + 105 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 570 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 78 T^{2} + 357 T^{3} + 78 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1054 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 15 T + 45 T^{2} - 178 T^{3} + 45 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 330 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 188 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 132 T^{2} + 812 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 111 T^{2} - 336 T^{3} + 111 p T^{4} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 18 T + 3 p T^{2} - 2076 T^{3} + 3 p^{2} T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 30 T + 540 T^{2} + 5884 T^{3} + 540 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 240 T^{2} - 42 T^{3} + 240 p T^{4} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57903804755330278085686120643, −7.13293847158859665058159762284, −6.95021128066406575018386818887, −6.91180531439014393290397596808, −6.51682187012133619995871098416, −6.39627660620405636396276341757, −6.19777772369333874188741494647, −5.52413732595204961828150156329, −5.34243411241196843794585098630, −5.33836776008858491195687882768, −4.93024537863108372450675907273, −4.54262669443514684687881423607, −4.38708978677541798773271231370, −4.26130629395073760600138611872, −4.05286301669264640289908821516, −3.40004659683032747711520173151, −3.26923664992524289127276913062, −2.96821878579255373942144743848, −2.80531589626159858471454016181, −2.24101649693341321378231731761, −2.14720643789115590642555610189, −1.73301818428220222525637088564, −1.02892573901714393175515767759, −0.70402146679869695856265643182, −0.27062119024830590549752893189,
0.27062119024830590549752893189, 0.70402146679869695856265643182, 1.02892573901714393175515767759, 1.73301818428220222525637088564, 2.14720643789115590642555610189, 2.24101649693341321378231731761, 2.80531589626159858471454016181, 2.96821878579255373942144743848, 3.26923664992524289127276913062, 3.40004659683032747711520173151, 4.05286301669264640289908821516, 4.26130629395073760600138611872, 4.38708978677541798773271231370, 4.54262669443514684687881423607, 4.93024537863108372450675907273, 5.33836776008858491195687882768, 5.34243411241196843794585098630, 5.52413732595204961828150156329, 6.19777772369333874188741494647, 6.39627660620405636396276341757, 6.51682187012133619995871098416, 6.91180531439014393290397596808, 6.95021128066406575018386818887, 7.13293847158859665058159762284, 7.57903804755330278085686120643
