| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·5-s + 4·6-s + 2·8-s + 9-s + 8·10-s + 4·11-s + 6·12-s − 16·13-s + 8·15-s + 4·17-s + 2·18-s + 12·20-s + 8·22-s + 4·23-s + 4·24-s + 12·25-s − 32·26-s − 2·27-s − 16·29-s + 16·30-s − 8·31-s − 6·32-s + 8·33-s + 8·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.78·5-s + 1.63·6-s + 0.707·8-s + 1/3·9-s + 2.52·10-s + 1.20·11-s + 1.73·12-s − 4.43·13-s + 2.06·15-s + 0.970·17-s + 0.471·18-s + 2.68·20-s + 1.70·22-s + 0.834·23-s + 0.816·24-s + 12/5·25-s − 6.27·26-s − 0.384·27-s − 2.97·29-s + 2.92·30-s − 1.43·31-s − 1.06·32-s + 1.39·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.864448153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.864448153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 56 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 112 T^{3} - 573 T^{4} + 112 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} + 64 T^{3} + 1955 T^{4} + 64 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 64 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 16 T + 88 T^{2} - 736 T^{3} + 8887 T^{4} - 736 p T^{5} + 88 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 102 T^{2} + 5915 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T + 656 T^{3} - 5905 T^{4} + 656 p T^{5} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 1640 p T^{5} + 140 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 208 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.762587299902956233208971903441, −9.311656244740961829791763617556, −9.078299705054228347854104539727, −9.035351245279866829599741745252, −8.727657081669744181563898226769, −8.136709774759497825112418332544, −7.61294315908616873244959779219, −7.57433717323487988304552963124, −7.32111794603211418314867556170, −6.94583933269027828633817373627, −6.84111233223846220194210314734, −6.55030564562380442777133054919, −5.80647170531078770796941100252, −5.57176317418056938817905304053, −5.56402701137154873391531483432, −5.35204718184660538430895470686, −4.69307314737029500098423310459, −4.57043138743411482840729084786, −4.17526723950096684770236833405, −3.43581297509463027675472343068, −3.11633590144782128325846682748, −3.03684330363718662259261721206, −2.30619245105572922245043296140, −1.98677109128099668675269595975, −1.93060194382655682929945983018,
1.93060194382655682929945983018, 1.98677109128099668675269595975, 2.30619245105572922245043296140, 3.03684330363718662259261721206, 3.11633590144782128325846682748, 3.43581297509463027675472343068, 4.17526723950096684770236833405, 4.57043138743411482840729084786, 4.69307314737029500098423310459, 5.35204718184660538430895470686, 5.56402701137154873391531483432, 5.57176317418056938817905304053, 5.80647170531078770796941100252, 6.55030564562380442777133054919, 6.84111233223846220194210314734, 6.94583933269027828633817373627, 7.32111794603211418314867556170, 7.57433717323487988304552963124, 7.61294315908616873244959779219, 8.136709774759497825112418332544, 8.727657081669744181563898226769, 9.035351245279866829599741745252, 9.078299705054228347854104539727, 9.311656244740961829791763617556, 9.762587299902956233208971903441
