| 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
| \(L(1)\) |
\(\approx\) |
\(2.042802637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042802637\) |
| \(L(\frac{3}{2})\) |
|
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
−7.21659241526536338659973143156, −7.16804134354512696656933847132, −6.85954001395315968840633825254, −6.67968991475679498652681377882, −6.31627276940851103054417792078, −6.14439516316244049904216342482, −6.13745720685053096602529873476, −5.55093024028473811839734572095, −5.51689194285687578038908052426, −5.35510555476023845310606569451, −5.30317302911706343136487690206, −4.55078410350496583271100922942, −4.33023319422812842324714389924, −4.24554910845157494204678499027, −4.19347482947790217823711413907, −3.90389022531421251169546721583, −3.37870933835700006399686548354, −2.60198949879693018158301370058, −2.52829665064565133915829300181, −2.38090728928741432852148037228, −2.29561502482684249903403017499, −1.82363596434009132830462614586, −1.19143020140226961146844775396, −0.969623683875399200582380364888, −0.50433916352877190701485076681,
0.50433916352877190701485076681, 0.969623683875399200582380364888, 1.19143020140226961146844775396, 1.82363596434009132830462614586, 2.29561502482684249903403017499, 2.38090728928741432852148037228, 2.52829665064565133915829300181, 2.60198949879693018158301370058, 3.37870933835700006399686548354, 3.90389022531421251169546721583, 4.19347482947790217823711413907, 4.24554910845157494204678499027, 4.33023319422812842324714389924, 4.55078410350496583271100922942, 5.30317302911706343136487690206, 5.35510555476023845310606569451, 5.51689194285687578038908052426, 5.55093024028473811839734572095, 6.13745720685053096602529873476, 6.14439516316244049904216342482, 6.31627276940851103054417792078, 6.67968991475679498652681377882, 6.85954001395315968840633825254, 7.16804134354512696656933847132, 7.21659241526536338659973143156
