| L(s) = 1 | + 6·5-s − 8·7-s + 18·11-s + 30·17-s − 6·19-s + 30·23-s − 5·25-s − 48·29-s + 42·31-s − 48·35-s − 62·37-s + 8·43-s + 174·47-s + 22·49-s + 78·53-s + 108·55-s − 78·59-s − 42·61-s + 58·67-s − 24·71-s + 318·73-s − 144·77-s − 110·79-s + 180·85-s + 378·89-s − 36·95-s + 54·101-s + ⋯ |
| L(s) = 1 | + 6/5·5-s − 8/7·7-s + 1.63·11-s + 1.76·17-s − 0.315·19-s + 1.30·23-s − 1/5·25-s − 1.65·29-s + 1.35·31-s − 1.37·35-s − 1.67·37-s + 8/43·43-s + 3.70·47-s + 0.448·49-s + 1.47·53-s + 1.96·55-s − 1.32·59-s − 0.688·61-s + 0.865·67-s − 0.338·71-s + 4.35·73-s − 1.87·77-s − 1.39·79-s + 2.11·85-s + 4.24·89-s − 0.378·95-s + 0.534·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.662890137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.662890137\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 8 T + 6 p T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 18 T + 19 T^{2} - 1134 T^{3} + 39180 T^{4} - 1134 p^{2} T^{5} + 19 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 30 T + 929 T^{2} - 1110 p T^{3} + 1380 p^{2} T^{4} - 1110 p^{3} T^{5} + 929 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 731 T^{2} + 4314 T^{3} + 390972 T^{4} + 4314 p^{2} T^{5} + 731 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 30 T - 221 T^{2} - 1890 T^{3} + 500700 T^{4} - 1890 p^{2} T^{5} - 221 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 24 T + 1754 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 42 T + 1307 T^{2} - 30198 T^{3} + 158508 T^{4} - 30198 p^{2} T^{5} + 1307 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 62 T + 1297 T^{2} - 11842 T^{3} - 649388 T^{4} - 11842 p^{2} T^{5} + 1297 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 3630 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 174 T + 17027 T^{2} - 1206690 T^{3} + 65507772 T^{4} - 1206690 p^{2} T^{5} + 17027 p^{4} T^{6} - 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 78 T - 767 T^{2} - 96174 T^{3} + 21955764 T^{4} - 96174 p^{2} T^{5} - 767 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 78 T + 5747 T^{2} + 290082 T^{3} + 8773068 T^{4} + 290082 p^{2} T^{5} + 5747 p^{4} T^{6} + 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 42 T + 2033 T^{2} + 60690 T^{3} - 9569868 T^{4} + 60690 p^{2} T^{5} + 2033 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 58 T - 2405 T^{2} + 186122 T^{3} - 1970756 T^{4} + 186122 p^{2} T^{5} - 2405 p^{4} T^{6} - 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 8318 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 318 T + 51257 T^{2} - 5580582 T^{3} + 459199092 T^{4} - 5580582 p^{2} T^{5} + 51257 p^{4} T^{6} - 318 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 110 T - 2957 T^{2} + 283250 T^{3} + 112247068 T^{4} + 283250 p^{2} T^{5} - 2957 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 378 T + 71921 T^{2} - 9182754 T^{3} + 904668996 T^{4} - 9182754 p^{2} T^{5} + 71921 p^{4} T^{6} - 378 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 26620 T^{2} + 330657414 T^{4} - 26620 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.98103522440337654470742554272, −6.67570384973563809576387158447, −6.30467505478017197518704962973, −6.29731304893548397289459594643, −6.16529502360746476967197292153, −5.67838231006937708985353107238, −5.60347059613379484159160462535, −5.56984056293077494805925301080, −5.19734247241271817370178921918, −4.81987667928115259172383779581, −4.71681698635789812604922226708, −4.36415230549445760215872785226, −3.81455847275747412887885291155, −3.79668318118551580178035468472, −3.59992664535128905924009532144, −3.48132524907413551582304197871, −3.10651560602821222286493485027, −2.59305724571531226616796798388, −2.37643285812709169102630887677, −2.29757963309829066841032156430, −1.78975212288291092939756399313, −1.34612373525956521621504541389, −1.17821184968360762133237485820, −0.73729641023663515143447214637, −0.43932548755545809137213541239,
0.43932548755545809137213541239, 0.73729641023663515143447214637, 1.17821184968360762133237485820, 1.34612373525956521621504541389, 1.78975212288291092939756399313, 2.29757963309829066841032156430, 2.37643285812709169102630887677, 2.59305724571531226616796798388, 3.10651560602821222286493485027, 3.48132524907413551582304197871, 3.59992664535128905924009532144, 3.79668318118551580178035468472, 3.81455847275747412887885291155, 4.36415230549445760215872785226, 4.71681698635789812604922226708, 4.81987667928115259172383779581, 5.19734247241271817370178921918, 5.56984056293077494805925301080, 5.60347059613379484159160462535, 5.67838231006937708985353107238, 6.16529502360746476967197292153, 6.29731304893548397289459594643, 6.30467505478017197518704962973, 6.67570384973563809576387158447, 6.98103522440337654470742554272
