| L(s) = 1 | − 486·3-s − 5.30e3·5-s − 3.88e4·7-s + 1.77e5·9-s + 1.04e6·11-s + 2.29e4·13-s + 2.57e6·15-s + 8.73e6·17-s − 7.34e6·19-s + 1.88e7·21-s + 6.71e6·23-s − 3.62e7·25-s − 5.73e7·27-s − 1.80e8·29-s + 2.11e8·31-s − 5.09e8·33-s + 2.06e8·35-s − 1.12e8·37-s − 1.11e7·39-s − 7.26e8·41-s − 2.16e8·43-s − 9.38e8·45-s − 2.17e9·47-s − 2.05e7·49-s − 4.24e9·51-s + 1.12e8·53-s − 5.55e9·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.758·5-s − 0.874·7-s + 9-s + 1.96·11-s + 0.0171·13-s + 0.875·15-s + 1.49·17-s − 0.680·19-s + 1.00·21-s + 0.217·23-s − 0.742·25-s − 0.769·27-s − 1.63·29-s + 1.32·31-s − 2.26·33-s + 0.663·35-s − 0.266·37-s − 0.0197·39-s − 0.979·41-s − 0.224·43-s − 0.758·45-s − 1.38·47-s − 0.0104·49-s − 1.72·51-s + 0.0367·53-s − 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.720413463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.720413463\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 212 p^{2} T + 12869446 p T^{2} + 212 p^{13} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 38872 T + 1531611582 T^{2} + 38872 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 1047400 T + 582443349302 T^{2} - 1047400 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22900 T + 3394284292494 T^{2} - 22900 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8733300 T + 76053146135686 T^{2} - 8733300 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7346600 T + 234273365886438 T^{2} + 7346600 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6711744 T + 1736184979908238 T^{2} - 6711744 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 180180692 T + 29060203717011374 T^{2} + 180180692 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 211581400 T + 49676382376667982 T^{2} - 211581400 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 112222700 T - 19046823489470754 T^{2} + 112222700 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 726961180 T + 1220821391254444982 T^{2} + 726961180 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 216408856 T + 301094542061252598 T^{2} + 216408856 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2174779088 T + 4972780192325352542 T^{2} + 2174779088 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 112024700 T + 2647443937277883614 T^{2} - 112024700 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3243949400 T + 1167855931192926038 T^{2} - 3243949400 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16526230620 T + \)\(15\!\cdots\!22\)\( T^{2} + 16526230620 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20772619112 T + \)\(26\!\cdots\!02\)\( T^{2} - 20772619112 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20637101600 T + \)\(45\!\cdots\!62\)\( T^{2} + 20637101600 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18548203500 T + \)\(70\!\cdots\!34\)\( T^{2} + 18548203500 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 28230083800 T + \)\(16\!\cdots\!78\)\( T^{2} - 28230083800 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7189282056 T - \)\(10\!\cdots\!82\)\( T^{2} + 7189282056 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 103679180788 T + \)\(82\!\cdots\!14\)\( T^{2} - 103679180788 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 199614486500 T + \)\(24\!\cdots\!26\)\( T^{2} - 199614486500 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.75558710091208521388889241922, −10.38711931289166725582153112903, −9.667674455658003750600482844651, −9.541929072384458588814988772652, −8.930596242072278805210475220225, −8.160973195302970014704941507944, −7.74233538783748232542033015403, −7.10651079472219361288077332414, −6.51694546260931046386083193221, −6.38807673981631125649405996952, −5.72513088429568138820312966132, −5.21874280124995312518767733386, −4.34760559758238238816118454706, −4.12309074808893218740617763550, −3.30273612823075766047577550029, −3.25612947615496204102997666376, −1.69780769014942430561537017542, −1.66804518455954627134452449803, −0.60758125067596832296851414466, −0.47566390401575337400700105261,
0.47566390401575337400700105261, 0.60758125067596832296851414466, 1.66804518455954627134452449803, 1.69780769014942430561537017542, 3.25612947615496204102997666376, 3.30273612823075766047577550029, 4.12309074808893218740617763550, 4.34760559758238238816118454706, 5.21874280124995312518767733386, 5.72513088429568138820312966132, 6.38807673981631125649405996952, 6.51694546260931046386083193221, 7.10651079472219361288077332414, 7.74233538783748232542033015403, 8.160973195302970014704941507944, 8.930596242072278805210475220225, 9.541929072384458588814988772652, 9.667674455658003750600482844651, 10.38711931289166725582153112903, 10.75558710091208521388889241922
