| L(s) = 1 | − 4·2-s + 8·4-s + 12·5-s + 10·7-s − 8·8-s + 6·9-s − 48·10-s + 36·11-s − 40·14-s − 4·16-s − 24·18-s + 52·19-s + 96·20-s − 144·22-s + 72·25-s + 80·28-s − 106·31-s + 32·32-s + 120·35-s + 48·36-s + 20·37-s − 208·38-s − 96·40-s − 24·41-s + 288·44-s + 72·45-s − 132·47-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s + 12/5·5-s + 10/7·7-s − 8-s + 2/3·9-s − 4.79·10-s + 3.27·11-s − 2.85·14-s − 1/4·16-s − 4/3·18-s + 2.73·19-s + 24/5·20-s − 6.54·22-s + 2.87·25-s + 20/7·28-s − 3.41·31-s + 32-s + 24/7·35-s + 4/3·36-s + 0.540·37-s − 5.47·38-s − 2.39·40-s − 0.585·41-s + 6.54·44-s + 8/5·45-s − 2.80·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.04555393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.04555393\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2594 T^{4} - 444 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 10 T + 50 T^{2} - 240 T^{3} + 527 T^{4} - 240 p^{2} T^{5} + 50 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 36 T + 648 T^{2} - 9972 T^{3} + 129122 T^{4} - 9972 p^{2} T^{5} + 648 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 772 T^{2} + 288390 T^{4} - 772 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T + 1352 T^{2} - 25116 T^{3} + 451694 T^{4} - 25116 p^{2} T^{5} + 1352 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1084 T^{2} + 604614 T^{4} - 1084 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 1382 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 106 T + 5618 T^{2} + 246768 T^{3} + 8970479 T^{4} + 246768 p^{2} T^{5} + 5618 p^{4} T^{6} + 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} + 10500 T^{3} - 3035986 T^{4} + 10500 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 13848 T^{3} - 552958 T^{4} + 13848 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6094 T^{2} + 15949011 T^{4} - 6094 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 132 T + 8712 T^{2} + 400884 T^{3} + 17761154 T^{4} + 400884 p^{2} T^{5} + 8712 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 36 T - 406 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T + 16200 T^{2} - 1354500 T^{3} + 96897314 T^{4} - 1354500 p^{2} T^{5} + 16200 p^{4} T^{6} - 180 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 72 T + 6863 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 38 T + 722 T^{2} + 72048 T^{3} + 465983 T^{4} + 72048 p^{2} T^{5} + 722 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T + 25992 T^{2} - 2465364 T^{3} + 200525954 T^{4} - 2465364 p^{2} T^{5} + 25992 p^{4} T^{6} - 228 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 190 T + 18050 T^{2} - 1719120 T^{3} + 149901647 T^{4} - 1719120 p^{2} T^{5} + 18050 p^{4} T^{6} - 190 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 48 T + 12911 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 264 T + 34848 T^{2} + 3047880 T^{3} + 244895714 T^{4} + 3047880 p^{2} T^{5} + 34848 p^{4} T^{6} + 264 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 336 T + 56448 T^{2} - 6596688 T^{3} + 633738434 T^{4} - 6596688 p^{2} T^{5} + 56448 p^{4} T^{6} - 336 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 p T + 2 p^{2} T^{2} - 13104 p T^{3} + 77694959 T^{4} - 13104 p^{3} T^{5} + 2 p^{6} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} \) |
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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
−6.90497570561933363498398952433, −6.78352667945824882965298744762, −6.61373998841638400408864240095, −6.36119307299258339406625896920, −6.32789033615146278952566728580, −5.61047054550105183373096548613, −5.53350395874125104491748727058, −5.53246281062637806100895719660, −5.04517162102751305972312593530, −4.99960001746685316627693558055, −4.99883567721763212216075279823, −4.16904085749451853741110990517, −3.96940858172753218863432808233, −3.83964505600091936448453797234, −3.69375341744616813034194086961, −3.16562801958643665553331441587, −3.05004990556356021220342301709, −2.18549038504650819548938580336, −2.10423878493959468922047560472, −1.86045446992436044702060833195, −1.85599317022744716442156756986, −1.39412735922902937860612946055, −1.05478959771780063357192154388, −0.864677402408173691937137951705, −0.68631695313620981812440112279,
0.68631695313620981812440112279, 0.864677402408173691937137951705, 1.05478959771780063357192154388, 1.39412735922902937860612946055, 1.85599317022744716442156756986, 1.86045446992436044702060833195, 2.10423878493959468922047560472, 2.18549038504650819548938580336, 3.05004990556356021220342301709, 3.16562801958643665553331441587, 3.69375341744616813034194086961, 3.83964505600091936448453797234, 3.96940858172753218863432808233, 4.16904085749451853741110990517, 4.99883567721763212216075279823, 4.99960001746685316627693558055, 5.04517162102751305972312593530, 5.53246281062637806100895719660, 5.53350395874125104491748727058, 5.61047054550105183373096548613, 6.32789033615146278952566728580, 6.36119307299258339406625896920, 6.61373998841638400408864240095, 6.78352667945824882965298744762, 6.90497570561933363498398952433
