| L(s) = 1 | + 2-s − 2·3-s + 2·4-s + 2·5-s − 2·6-s − 7-s + 3·9-s + 2·10-s − 4·12-s + 4·13-s − 14-s − 4·15-s + 4·17-s + 3·18-s + 4·20-s + 2·21-s − 16·23-s + 5·25-s + 4·26-s − 2·28-s − 6·29-s − 4·30-s − 10·31-s − 11·32-s + 4·34-s − 2·35-s + 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 4-s + 0.894·5-s − 0.816·6-s − 0.377·7-s + 9-s + 0.632·10-s − 1.15·12-s + 1.10·13-s − 0.267·14-s − 1.03·15-s + 0.970·17-s + 0.707·18-s + 0.894·20-s + 0.436·21-s − 3.33·23-s + 25-s + 0.784·26-s − 0.377·28-s − 1.11·29-s − 0.730·30-s − 1.79·31-s − 1.94·32-s + 0.685·34-s − 0.338·35-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4258389571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4258389571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 4 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 12 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 40 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 4 T - T^{2} + 72 T^{3} - 271 T^{4} + 72 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 10 T + 69 T^{2} + 380 T^{3} + 1661 T^{4} + 380 p T^{5} + 69 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 228 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 4 T - 25 T^{2} + 264 T^{3} - 31 T^{4} + 264 p T^{5} - 25 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - 10 T + 53 T^{2} - 60 T^{3} - 1891 T^{4} - 60 p T^{5} + 53 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 2 T - 55 T^{2} - 228 T^{3} + 2789 T^{4} - 228 p T^{5} - 55 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 24 T^{3} - 4895 T^{4} - 24 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 8 T - 9 T^{2} - 656 T^{3} - 4591 T^{4} - 656 p T^{5} - 9 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 752 p T^{5} - 15 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 10 T + 3 T^{2} + 940 T^{3} - 9691 T^{4} + 940 p T^{5} + 3 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.26307489520927824418314684200, −6.83468089114749104081318642101, −6.75684787467917635644124362813, −6.48738697640217868571427883917, −6.44529307839690765626180767451, −6.14080678657937278241661702766, −5.74407884764495685934017929081, −5.70370365490755499262595809528, −5.69196907920081428697993298372, −5.35594289706338941167640462441, −4.94510716776241660904599911135, −4.83336400590443012769423252647, −4.75069485523418102033865767277, −4.02234352457002809873634029249, −3.66357656518717343341756590537, −3.63521162200154162409984407459, −3.59008820230332564870684767909, −3.42572928598232855003846395413, −2.70022992069080578136801578278, −2.36046672107023033874017675541, −2.00648132285814175127456105490, −1.97682126617812537798111735642, −1.37078141799585361891768254651, −1.22171845552555633974236982378, −0.14191867392086788064135270729,
0.14191867392086788064135270729, 1.22171845552555633974236982378, 1.37078141799585361891768254651, 1.97682126617812537798111735642, 2.00648132285814175127456105490, 2.36046672107023033874017675541, 2.70022992069080578136801578278, 3.42572928598232855003846395413, 3.59008820230332564870684767909, 3.63521162200154162409984407459, 3.66357656518717343341756590537, 4.02234352457002809873634029249, 4.75069485523418102033865767277, 4.83336400590443012769423252647, 4.94510716776241660904599911135, 5.35594289706338941167640462441, 5.69196907920081428697993298372, 5.70370365490755499262595809528, 5.74407884764495685934017929081, 6.14080678657937278241661702766, 6.44529307839690765626180767451, 6.48738697640217868571427883917, 6.75684787467917635644124362813, 6.83468089114749104081318642101, 7.26307489520927824418314684200
