| L(s) = 1 | + 2-s − 3-s + 5-s − 6-s + 2·8-s + 9-s + 10-s + 11-s + 4·13-s − 15-s + 3·16-s + 10·17-s + 18-s + 22-s + 10·23-s − 2·24-s + 25-s + 4·26-s + 6·27-s − 2·29-s − 30-s − 18·31-s + 3·32-s − 33-s + 10·34-s − 5·37-s − 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s + 0.707·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 1.10·13-s − 0.258·15-s + 3/4·16-s + 2.42·17-s + 0.235·18-s + 0.213·22-s + 2.08·23-s − 0.408·24-s + 1/5·25-s + 0.784·26-s + 1.15·27-s − 0.371·29-s − 0.182·30-s − 3.23·31-s + 0.530·32-s − 0.174·33-s + 1.71·34-s − 0.821·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.592576868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.592576868\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + T - 7 T^{3} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 3 p T^{3} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 20 T^{2} + T^{3} + 20 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 100 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 63 T^{2} - 300 T^{3} + 63 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 41 T^{2} - 16 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 3 p T^{2} - 312 T^{3} + 3 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} - 68 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 66 T^{2} + 281 T^{3} + 66 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 102 T^{2} + 111 T^{3} + 102 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 36 T^{2} + 61 T^{3} + 36 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 15 T + 194 T^{2} + 1439 T^{3} + 194 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 36 T^{2} + 221 T^{3} + 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 12 T + 87 T^{2} + 520 T^{3} + 87 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 257 T^{2} - 1944 T^{3} + 257 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 101 T^{2} + 520 T^{3} + 101 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 15 T + 264 T^{2} + 2167 T^{3} + 264 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 23 T + 404 T^{2} - 4111 T^{3} + 404 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 12 T + 195 T^{2} - 1736 T^{3} + 195 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 159 T^{2} - 1316 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.948419771901176393901459103847, −7.66538722156983502864754925710, −7.52080608376219938747136763177, −7.11808548403691056536603939639, −6.87963941678374295969396274861, −6.74838905849277702458037657862, −6.34085296793394346458087015207, −5.98947012292366807240591761222, −5.76342100682712222720009883002, −5.69003365190174944137172582824, −5.16463718678231989928856621030, −5.10491486699606127817765904391, −4.88223441833822998433189519596, −4.57836327824548989353574849969, −4.18968255550360709859192707918, −3.77424368455754523772619267819, −3.45279655274750026068710980764, −3.37314697773874377190699614916, −3.18809471073647067614764595629, −2.53573116649290944582137830649, −2.22696758870856170620768087803, −1.41426624750961010284880474197, −1.33268930215947956044915666917, −1.31337814247338217983810923633, −0.49343659536183365737855050038,
0.49343659536183365737855050038, 1.31337814247338217983810923633, 1.33268930215947956044915666917, 1.41426624750961010284880474197, 2.22696758870856170620768087803, 2.53573116649290944582137830649, 3.18809471073647067614764595629, 3.37314697773874377190699614916, 3.45279655274750026068710980764, 3.77424368455754523772619267819, 4.18968255550360709859192707918, 4.57836327824548989353574849969, 4.88223441833822998433189519596, 5.10491486699606127817765904391, 5.16463718678231989928856621030, 5.69003365190174944137172582824, 5.76342100682712222720009883002, 5.98947012292366807240591761222, 6.34085296793394346458087015207, 6.74838905849277702458037657862, 6.87963941678374295969396274861, 7.11808548403691056536603939639, 7.52080608376219938747136763177, 7.66538722156983502864754925710, 7.948419771901176393901459103847
