| L(s) = 1 | + 4·5-s − 4·7-s − 8·11-s + 4·13-s + 8·19-s + 12·23-s + 10·25-s + 4·29-s + 16·31-s − 16·35-s + 4·37-s + 12·41-s − 16·43-s + 8·49-s − 4·53-s − 32·55-s − 16·59-s + 4·61-s + 16·65-s + 8·67-s − 12·71-s + 28·73-s + 32·77-s + 16·83-s − 12·89-s − 16·91-s + 32·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s − 2.41·11-s + 1.10·13-s + 1.83·19-s + 2.50·23-s + 2·25-s + 0.742·29-s + 2.87·31-s − 2.70·35-s + 0.657·37-s + 1.87·41-s − 2.43·43-s + 8/7·49-s − 0.549·53-s − 4.31·55-s − 2.08·59-s + 0.512·61-s + 1.98·65-s + 0.977·67-s − 1.42·71-s + 3.27·73-s + 3.64·77-s + 1.75·83-s − 1.27·89-s − 1.67·91-s + 3.28·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.164529930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.164529930\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 160 T^{3} - 1246 T^{4} + 160 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 204 T^{3} - 830 T^{4} + 204 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 3694 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 162 T^{2} + 1384 T^{3} + 10178 T^{4} + 1384 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 54 T^{2} + 708 T^{3} + 3490 T^{4} + 708 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 16 T + 114 T^{2} + 696 T^{3} + 4834 T^{4} + 696 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 736 T^{3} - 5854 T^{4} + 736 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 876 T^{3} + 10654 T^{4} + 876 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 114 T^{2} - 792 T^{3} + 6370 T^{4} - 792 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.92613936415673743160375247050, −6.64078033840874007841303405008, −6.63834548714558045257441208317, −6.26578050801755873565194577054, −6.24504914295008594772992172111, −5.88871003924765065431391411118, −5.56083427978972884646479591167, −5.50691490082748807937135468321, −5.31937647296716570949981876289, −4.94544858219513246933934420331, −4.85127007796923068943460270401, −4.72819062680994929194977626480, −4.34624790179702755383236326931, −3.91728883661682922292029970341, −3.44236516932480023170301866659, −3.43015518856683004835495000622, −3.07678376222119555791612015242, −2.74990769601051733156194435853, −2.64751007971841239694284996515, −2.54264168329440708203481312112, −2.31620501130018520575498621490, −1.29742766697671368267933058533, −1.24753920055481025693692017910, −1.18706530160206775151018304693, −0.38613089640302038840131141817,
0.38613089640302038840131141817, 1.18706530160206775151018304693, 1.24753920055481025693692017910, 1.29742766697671368267933058533, 2.31620501130018520575498621490, 2.54264168329440708203481312112, 2.64751007971841239694284996515, 2.74990769601051733156194435853, 3.07678376222119555791612015242, 3.43015518856683004835495000622, 3.44236516932480023170301866659, 3.91728883661682922292029970341, 4.34624790179702755383236326931, 4.72819062680994929194977626480, 4.85127007796923068943460270401, 4.94544858219513246933934420331, 5.31937647296716570949981876289, 5.50691490082748807937135468321, 5.56083427978972884646479591167, 5.88871003924765065431391411118, 6.24504914295008594772992172111, 6.26578050801755873565194577054, 6.63834548714558045257441208317, 6.64078033840874007841303405008, 6.92613936415673743160375247050
