| L(s) = 1 | − 2·2-s + 2·3-s + 4-s + 4·5-s − 4·6-s − 2·7-s + 2·8-s + 5·9-s − 8·10-s − 2·11-s + 2·12-s − 4·13-s + 4·14-s + 8·15-s − 4·16-s + 4·17-s − 10·18-s + 4·19-s + 4·20-s − 4·21-s + 4·22-s + 4·23-s + 4·24-s + 12·25-s + 8·26-s + 10·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.78·5-s − 1.63·6-s − 0.755·7-s + 0.707·8-s + 5/3·9-s − 2.52·10-s − 0.603·11-s + 0.577·12-s − 1.10·13-s + 1.06·14-s + 2.06·15-s − 16-s + 0.970·17-s − 2.35·18-s + 0.917·19-s + 0.894·20-s − 0.872·21-s + 0.852·22-s + 0.834·23-s + 0.816·24-s + 12/5·25-s + 1.56·26-s + 1.92·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.193501392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.193501392\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4\times C_2$ | \( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 112 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 24 T^{2} - 8 T^{3} + 935 T^{4} - 8 p T^{5} - 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 16 T^{2} + 56 T^{3} + 127 T^{4} + 56 p T^{5} - 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 16 T + 120 T^{2} - 992 T^{3} + 7655 T^{4} - 992 p T^{5} + 120 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 10 T^{2} + 272 T^{3} - 2285 T^{4} + 272 p T^{5} - 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} + 376 T^{3} - 3513 T^{4} + 376 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 14 T + 31 T^{2} - 658 T^{3} + 12180 T^{4} - 658 p T^{5} + 31 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 129 T^{2} + 1314 T^{3} + 14540 T^{4} + 1314 p T^{5} + 129 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 434 p T^{5} + 31 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 48 T^{2} - 992 T^{3} + 19247 T^{4} - 992 p T^{5} + 48 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 103 T^{2} - 1134 T^{3} + 17004 T^{4} - 1134 p T^{5} + 103 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 448 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.605279747146972834999705252091, −9.461100734723678984606578818064, −9.004304191124823614415120313252, −8.967848038800143092760677460070, −8.463570247834292851715321678827, −8.299257505895437554564725469101, −7.76059177999425684782929934578, −7.63430176786252485910067063432, −7.62484930781863561527086030701, −7.04878483542018851296720419184, −6.75083135928055445881585621403, −6.68205235612353318408213383351, −6.22983899953747251445802437981, −5.60298148705293741684887042122, −5.44871199081206338730042416471, −5.31643281918903787146830163960, −4.60758829445336349660510693597, −4.59308254135372587184593459195, −3.89170027425763571965200180972, −3.51784443134322464881269364834, −2.83541669813597376153202499221, −2.71524795317829369046764101724, −2.34699105183237197255238688742, −1.52392622082547512492442252022, −1.18793482470142238748542472840,
1.18793482470142238748542472840, 1.52392622082547512492442252022, 2.34699105183237197255238688742, 2.71524795317829369046764101724, 2.83541669813597376153202499221, 3.51784443134322464881269364834, 3.89170027425763571965200180972, 4.59308254135372587184593459195, 4.60758829445336349660510693597, 5.31643281918903787146830163960, 5.44871199081206338730042416471, 5.60298148705293741684887042122, 6.22983899953747251445802437981, 6.68205235612353318408213383351, 6.75083135928055445881585621403, 7.04878483542018851296720419184, 7.62484930781863561527086030701, 7.63430176786252485910067063432, 7.76059177999425684782929934578, 8.299257505895437554564725469101, 8.463570247834292851715321678827, 8.967848038800143092760677460070, 9.004304191124823614415120313252, 9.461100734723678984606578818064, 9.605279747146972834999705252091
