| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·5-s + 4·6-s + 2·8-s + 3·9-s − 8·10-s + 8·11-s − 2·12-s − 8·15-s − 4·16-s − 4·17-s − 6·18-s − 10·19-s + 4·20-s − 16·22-s − 4·23-s − 4·24-s + 2·25-s − 10·27-s + 4·29-s + 16·30-s − 12·31-s + 2·32-s − 16·33-s + 8·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.78·5-s + 1.63·6-s + 0.707·8-s + 9-s − 2.52·10-s + 2.41·11-s − 0.577·12-s − 2.06·15-s − 16-s − 0.970·17-s − 1.41·18-s − 2.29·19-s + 0.894·20-s − 3.41·22-s − 0.834·23-s − 0.816·24-s + 2/5·25-s − 1.92·27-s + 0.742·29-s + 2.92·30-s − 2.15·31-s + 0.353·32-s − 2.78·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5310102129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5310102129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 43 T^{2} + 10 p T^{3} + 52 p T^{4} + 10 p^{2} T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 80 p T^{5} - 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 34 T^{2} - 368 T^{3} - 1637 T^{4} - 368 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 26 T^{2} - 144 T^{3} + 3483 T^{4} - 144 p T^{5} + 26 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 1350 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 640 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 110 T^{2} + 80 T^{3} + 9379 T^{4} + 80 p T^{5} - 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 90 p T^{5} - 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 2880 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 80 p T^{5} - 158 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.31795229838990267602411280007, −7.29341928002683619292016996429, −6.50931613486479518830314645520, −6.42370062594971783777739013410, −6.39304700975695755152412476065, −6.33506908815513487277905301737, −6.06213700572018074558351337221, −5.95513418106810129585405706977, −5.27184348085651030101805404264, −5.23580692433313369841440844780, −5.12274374405865674276061845719, −4.83200072007190599910677376256, −4.36909570985778036116497121638, −4.11194901763219405765772350613, −3.82924546225711341446846482843, −3.78746534202866095828487920908, −3.66795823457915723888988327994, −2.96311067988397857300236062475, −2.22888650173269534816578068575, −2.08123198358626318191127934493, −2.06846130084708446274122502104, −1.75287837712417436074424923068, −1.38100215914070963429302469757, −0.859749523137761223772017627818, −0.29157308999385689537668583585,
0.29157308999385689537668583585, 0.859749523137761223772017627818, 1.38100215914070963429302469757, 1.75287837712417436074424923068, 2.06846130084708446274122502104, 2.08123198358626318191127934493, 2.22888650173269534816578068575, 2.96311067988397857300236062475, 3.66795823457915723888988327994, 3.78746534202866095828487920908, 3.82924546225711341446846482843, 4.11194901763219405765772350613, 4.36909570985778036116497121638, 4.83200072007190599910677376256, 5.12274374405865674276061845719, 5.23580692433313369841440844780, 5.27184348085651030101805404264, 5.95513418106810129585405706977, 6.06213700572018074558351337221, 6.33506908815513487277905301737, 6.39304700975695755152412476065, 6.42370062594971783777739013410, 6.50931613486479518830314645520, 7.29341928002683619292016996429, 7.31795229838990267602411280007
