| L(s) = 1 | − 201·4-s + 872·13-s + 1.83e4·16-s + 1.95e3·19-s + 300·25-s − 7.51e5·31-s + 8.46e5·37-s − 5.15e5·43-s − 1.75e5·52-s + 3.62e6·61-s − 2.53e6·64-s − 6.94e6·67-s − 4.80e6·73-s − 3.92e5·76-s − 7.97e6·79-s − 2.64e7·97-s − 6.03e4·100-s − 5.15e7·103-s − 6.03e6·109-s + 2.05e7·121-s + 1.51e8·124-s + 127-s + 131-s + 137-s + 139-s − 1.70e8·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1.57·4-s + 0.110·13-s + 1.11·16-s + 0.0652·19-s + 0.00383·25-s − 4.53·31-s + 2.74·37-s − 0.989·43-s − 0.172·52-s + 2.04·61-s − 1.21·64-s − 2.82·67-s − 1.44·73-s − 0.102·76-s − 1.81·79-s − 2.93·97-s − 0.00602·100-s − 4.64·103-s − 0.446·109-s + 1.05·121-s + 7.11·124-s − 4.31·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 201 T^{2} + 5519 p^{2} T^{4} + 201 p^{14} T^{6} + p^{28} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 12 p^{2} T^{2} + 18200582 p^{4} T^{4} - 12 p^{16} T^{6} + p^{28} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 20517108 T^{2} + 742788783567782 T^{4} - 20517108 p^{14} T^{6} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 436 T + 113568222 T^{2} - 436 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 1184398628 T^{2} + 644094766406111478 T^{4} + 1184398628 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 976 T + 668859558 T^{2} - 976 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 2043936380 T^{2} + 4227870824516665302 T^{4} + 2043936380 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 61875985812 T^{2} + \)\(15\!\cdots\!54\)\( T^{4} + 61875985812 p^{14} T^{6} + p^{28} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 375792 T + 86304755438 T^{2} + 375792 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 423228 T + 184044219662 T^{2} - 423228 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 156281459460 T^{2} + \)\(24\!\cdots\!86\)\( T^{4} + 156281459460 p^{14} T^{6} + p^{28} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 257816 T - 192263940138 T^{2} + 257816 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 842134473404 T^{2} + \)\(34\!\cdots\!58\)\( T^{4} + 842134473404 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 3326473286132 T^{2} + \)\(51\!\cdots\!78\)\( T^{4} + 3326473286132 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 4456568532044 T^{2} + \)\(15\!\cdots\!42\)\( T^{4} + 4456568532044 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 1812220 T + 5867336171742 T^{2} - 1812220 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 3474184 T + 15112079511654 T^{2} + 3474184 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 25938591355580 T^{2} + \)\(32\!\cdots\!86\)\( T^{4} + 25938591355580 p^{14} T^{6} + p^{28} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 2402316 T + 16607904562262 T^{2} + 2402316 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 3985280 T + 18748061973534 T^{2} + 3985280 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 63798783848684 T^{2} + \)\(24\!\cdots\!78\)\( T^{4} + 63798783848684 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 10918057299780 T^{2} - \)\(26\!\cdots\!14\)\( T^{4} + 10918057299780 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 13209420 T + 199428343077542 T^{2} + 13209420 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−7.22290603344988336301986980323, −7.08926405520548166486837212828, −7.02758222948378392936511817411, −6.70236293960251087117146873865, −6.20241489733088860573660916378, −5.96315997290670617875388524465, −5.79575590545381325559169980282, −5.50763147359694253509773879462, −5.45561919954731262073390214278, −5.15714230849839361698783787593, −4.64595051292215362771311769696, −4.59290264574360834876121042783, −4.38557704550402545317358517414, −3.88978497812687482661261074873, −3.83210944461828898369979328831, −3.76802829717813821984359254145, −3.13415313170872516660259226493, −3.07471786634932777931437298874, −2.58440770941864225830799641661, −2.39513577544194063512907891847, −2.04043526468004908742248694077, −1.47103813228875450465115701989, −1.36914322767741741145165040910, −1.23355083761497584308059214194, −0.806449024477491149145703278725, 0, 0, 0, 0,
0.806449024477491149145703278725, 1.23355083761497584308059214194, 1.36914322767741741145165040910, 1.47103813228875450465115701989, 2.04043526468004908742248694077, 2.39513577544194063512907891847, 2.58440770941864225830799641661, 3.07471786634932777931437298874, 3.13415313170872516660259226493, 3.76802829717813821984359254145, 3.83210944461828898369979328831, 3.88978497812687482661261074873, 4.38557704550402545317358517414, 4.59290264574360834876121042783, 4.64595051292215362771311769696, 5.15714230849839361698783787593, 5.45561919954731262073390214278, 5.50763147359694253509773879462, 5.79575590545381325559169980282, 5.96315997290670617875388524465, 6.20241489733088860573660916378, 6.70236293960251087117146873865, 7.02758222948378392936511817411, 7.08926405520548166486837212828, 7.22290603344988336301986980323
