| L(s) = 1 | − 4·2-s + 3-s + 12·4-s − 3·5-s − 4·6-s − 7-s − 25·8-s + 12·10-s + 11-s + 12·12-s − 11·13-s + 4·14-s − 3·15-s + 45·16-s + 3·17-s − 4·19-s − 36·20-s − 21-s − 4·22-s − 20·23-s − 25·24-s + 10·25-s + 44·26-s − 12·28-s − 3·29-s + 12·30-s − 3·31-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 6·4-s − 1.34·5-s − 1.63·6-s − 0.377·7-s − 8.83·8-s + 3.79·10-s + 0.301·11-s + 3.46·12-s − 3.05·13-s + 1.06·14-s − 0.774·15-s + 45/4·16-s + 0.727·17-s − 0.917·19-s − 8.04·20-s − 0.218·21-s − 0.852·22-s − 4.17·23-s − 5.10·24-s + 2·25-s + 8.62·26-s − 2.26·28-s − 0.557·29-s + 2.19·30-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02378765081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02378765081\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + p^{2} T + p^{2} T^{2} - 7 T^{3} - 21 T^{4} - 7 p T^{5} + p^{4} T^{6} + p^{5} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 3 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 11 T + 48 T^{2} + 145 T^{3} + 491 T^{4} + 145 p T^{5} + 48 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 3 T - 8 T^{2} + 75 T^{3} - 89 T^{4} + 75 p T^{5} - 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 88 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 3 T - 20 T^{2} - 147 T^{3} + 139 T^{4} - 147 p T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 3 T - 12 T^{2} + 131 T^{3} + 1365 T^{4} + 131 p T^{5} - 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 6 T + 39 T^{2} + 352 T^{3} + 3309 T^{4} + 352 p T^{5} + 39 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 15 T + 59 T^{2} + 45 T^{3} + 256 T^{4} + 45 p T^{5} + 59 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 8 T + 67 T^{2} - 610 T^{3} + 6231 T^{4} - 610 p T^{5} + 67 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 11 T + 43 T^{2} + 565 T^{3} + 6936 T^{4} + 565 p T^{5} + 43 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 9 T + 77 T^{2} + 477 T^{3} + 700 T^{4} + 477 p T^{5} + 77 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - 15 T + 74 T^{2} - 675 T^{3} + 8491 T^{4} - 675 p T^{5} + 74 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 15 T + 44 T^{2} - 945 T^{3} - 13979 T^{4} - 945 p T^{5} + 44 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 7 T - 39 T^{2} + 161 T^{3} + 6764 T^{4} + 161 p T^{5} - 39 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 11 T - 33 T^{2} - 557 T^{3} + 80 T^{4} - 557 p T^{5} - 33 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 14 T + 193 T^{2} + 2140 T^{3} + 26421 T^{4} + 2140 p T^{5} + 193 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 21 T + 227 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 33 T + 452 T^{2} - 4035 T^{3} + 36031 T^{4} - 4035 p T^{5} + 452 p^{2} T^{6} - 33 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.973432883554362188689702707892, −8.331843907894732227599571819918, −8.301452084864150850446351078094, −8.185461155209004286643589092630, −7.993898932392597326638770060303, −7.82987854008509418057086213217, −7.27785477966797319823951118705, −7.12224687267854619582669226270, −7.06914598955019881708500646147, −7.01272608907639050146532316387, −6.36968832710382459965484475667, −6.27475659065205344952102446760, −5.95177059252378348657252491087, −5.31142028221977860744088455203, −5.02911834213882559099577330579, −4.67417751463471007293925104374, −4.36772479626581962474302252876, −3.50764289190871087620009182506, −3.50248675518077223148018888677, −3.26044887810231908010556120804, −2.56220895859971984361298126914, −2.06207235870521909400671977218, −1.80881972884943938460824519444, −1.80033546662995535019263700637, −0.11886784829226630252888362140,
0.11886784829226630252888362140, 1.80033546662995535019263700637, 1.80881972884943938460824519444, 2.06207235870521909400671977218, 2.56220895859971984361298126914, 3.26044887810231908010556120804, 3.50248675518077223148018888677, 3.50764289190871087620009182506, 4.36772479626581962474302252876, 4.67417751463471007293925104374, 5.02911834213882559099577330579, 5.31142028221977860744088455203, 5.95177059252378348657252491087, 6.27475659065205344952102446760, 6.36968832710382459965484475667, 7.01272608907639050146532316387, 7.06914598955019881708500646147, 7.12224687267854619582669226270, 7.27785477966797319823951118705, 7.82987854008509418057086213217, 7.993898932392597326638770060303, 8.185461155209004286643589092630, 8.301452084864150850446351078094, 8.331843907894732227599571819918, 8.973432883554362188689702707892
