| L(s) = 1 | − 14·5-s + 24·7-s + 32·11-s − 56·13-s − 154·17-s + 224·19-s + 68·23-s + 151·25-s + 472·29-s + 196·31-s − 336·35-s − 346·37-s + 840·41-s − 688·43-s + 84·47-s + 338·49-s − 438·53-s − 448·55-s − 56·59-s − 98·61-s + 784·65-s − 336·67-s − 1.79e3·71-s − 966·73-s + 768·77-s + 52·79-s − 784·83-s + ⋯ |
| L(s) = 1 | − 1.25·5-s + 1.29·7-s + 0.877·11-s − 1.19·13-s − 2.19·17-s + 2.70·19-s + 0.616·23-s + 1.20·25-s + 3.02·29-s + 1.13·31-s − 1.62·35-s − 1.53·37-s + 3.19·41-s − 2.43·43-s + 0.260·47-s + 0.985·49-s − 1.13·53-s − 1.09·55-s − 0.123·59-s − 0.205·61-s + 1.49·65-s − 0.612·67-s − 2.99·71-s − 1.54·73-s + 1.13·77-s + 0.0740·79-s − 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02275734201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02275734201\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 24 T + 34 p T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 14 T + 9 p T^{2} - 1386 T^{3} - 17324 T^{4} - 1386 p^{3} T^{5} + 9 p^{7} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 32 T - 81 T^{2} + 49824 T^{3} - 1913480 T^{4} + 49824 p^{3} T^{5} - 81 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 28 T + 3998 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 154 T + 8553 T^{2} + 821898 T^{3} + 89262292 T^{4} + 821898 p^{3} T^{5} + 8553 p^{6} T^{6} + 154 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 224 T + 24247 T^{2} - 2735264 T^{3} + 281109976 T^{4} - 2735264 p^{3} T^{5} + 24247 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 68 T - 19053 T^{2} + 44676 T^{3} + 356304232 T^{4} + 44676 p^{3} T^{5} - 19053 p^{6} T^{6} - 68 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 236 T + 33694 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 196 T - 28957 T^{2} - 1527036 T^{3} + 2507166392 T^{4} - 1527036 p^{3} T^{5} - 28957 p^{6} T^{6} - 196 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 346 T + 53749 T^{2} - 12227294 T^{3} - 4278055868 T^{4} - 12227294 p^{3} T^{5} + 53749 p^{6} T^{6} + 346 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 420 T + 176614 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 84 T - 194029 T^{2} + 551124 T^{3} + 28923386808 T^{4} + 551124 p^{3} T^{5} - 194029 p^{6} T^{6} - 84 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 438 T - 88603 T^{2} - 7580466 T^{3} + 27924999492 T^{4} - 7580466 p^{3} T^{5} - 88603 p^{6} T^{6} + 438 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 56 T - 357753 T^{2} - 2792664 T^{3} + 87416268136 T^{4} - 2792664 p^{3} T^{5} - 357753 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 98 T - 244147 T^{2} - 19620678 T^{3} + 10689270116 T^{4} - 19620678 p^{3} T^{5} - 244147 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 336 T - 515041 T^{2} + 8874096 T^{3} + 269891554152 T^{4} + 8874096 p^{3} T^{5} - 515041 p^{6} T^{6} + 336 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 896 T + 800494 T^{2} + 896 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 966 T - 63367 T^{2} + 39606 p^{2} T^{3} + 1230564 p^{3} T^{4} + 39606 p^{5} T^{5} - 63367 p^{6} T^{6} + 966 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 52 T - 329557 T^{2} + 33998484 T^{3} - 134023260856 T^{4} + 33998484 p^{3} T^{5} - 329557 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 392 T + 895462 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 294 T - 1211911 T^{2} + 32807754 T^{3} + 1127788940964 T^{4} + 32807754 p^{3} T^{5} - 1211911 p^{6} T^{6} - 294 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 420 T + 1655734 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.244732836903609162887837826654, −8.056155979452303324367411679704, −7.86706128157970638772533027491, −7.46161381670112291694113453459, −7.24589116711710948892923864963, −7.12835938537618783374176348263, −6.77687890553550452350249835517, −6.50409511627148242776915868371, −6.39474123395813983443199172590, −5.73712375630595232853011152918, −5.59114278170465606965992374181, −5.01703318759105533670282244546, −4.93999339069085495119706031777, −4.61302477518302056427244929586, −4.45483433597741805502673295044, −4.25880925932733670737552323964, −3.78418052550444808790840383036, −3.33630791522076643075068287972, −2.83898607292261422340164062979, −2.65320498276553849061212475572, −2.55981552287030904010371158090, −1.48054364312574821224525611490, −1.24593130288034270956631095977, −1.11828231992224251567048809897, −0.02804105749147295303977499782,
0.02804105749147295303977499782, 1.11828231992224251567048809897, 1.24593130288034270956631095977, 1.48054364312574821224525611490, 2.55981552287030904010371158090, 2.65320498276553849061212475572, 2.83898607292261422340164062979, 3.33630791522076643075068287972, 3.78418052550444808790840383036, 4.25880925932733670737552323964, 4.45483433597741805502673295044, 4.61302477518302056427244929586, 4.93999339069085495119706031777, 5.01703318759105533670282244546, 5.59114278170465606965992374181, 5.73712375630595232853011152918, 6.39474123395813983443199172590, 6.50409511627148242776915868371, 6.77687890553550452350249835517, 7.12835938537618783374176348263, 7.24589116711710948892923864963, 7.46161381670112291694113453459, 7.86706128157970638772533027491, 8.056155979452303324367411679704, 8.244732836903609162887837826654
