| L(s) = 1 | + 18·3-s − 10·5-s + 98·7-s + 243·9-s + 54·11-s + 240·13-s − 180·15-s + 1.90e3·17-s − 400·19-s + 1.76e3·21-s + 2.93e3·23-s − 1.89e3·25-s + 2.91e3·27-s + 5.54e3·29-s + 4.90e3·31-s + 972·33-s − 980·35-s + 2.95e3·37-s + 4.32e3·39-s + 6.07e3·41-s + 3.30e3·43-s − 2.43e3·45-s + 1.69e4·47-s + 7.20e3·49-s + 3.43e4·51-s + 6.89e3·53-s − 540·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.178·5-s + 0.755·7-s + 9-s + 0.134·11-s + 0.393·13-s − 0.206·15-s + 1.59·17-s − 0.254·19-s + 0.872·21-s + 1.15·23-s − 0.606·25-s + 0.769·27-s + 1.22·29-s + 0.916·31-s + 0.155·33-s − 0.135·35-s + 0.354·37-s + 0.454·39-s + 0.563·41-s + 0.272·43-s − 0.178·45-s + 1.12·47-s + 3/7·49-s + 1.84·51-s + 0.337·53-s − 0.0240·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.512824452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.512824452\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 p T + 1994 T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 54 T + 284302 T^{2} - 54 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 240 T + 739862 T^{2} - 240 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1906 T + 1860002 T^{2} - 1906 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 400 T + 3279798 T^{2} + 400 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2930 T + 9774686 T^{2} - 2930 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5540 T + 40407182 T^{2} - 5540 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4904 T + 59914302 T^{2} - 4904 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2952 T + 138794486 T^{2} - 2952 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6070 T + 232254602 T^{2} - 6070 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3304 T + 197837766 T^{2} - 3304 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16988 T + 429309854 T^{2} - 16988 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6896 T + 613869254 T^{2} - 6896 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 53820 T + 1699029142 T^{2} - 53820 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 68 p T + 1287381294 T^{2} - 68 p^{6} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 33516 T + 2261444678 T^{2} - 33516 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 73518 T + 4934300734 T^{2} - 73518 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 128 T - 360358674 T^{2} - 128 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 57740 T + 6120687198 T^{2} - 57740 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 23016 T - 4303421546 T^{2} - 23016 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 141530 T + 14809201898 T^{2} + 141530 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 226216 T + 29691907182 T^{2} + 226216 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.98547328557299080220137017791, −11.92387253696974456396438528333, −10.93195591872115091424482044338, −10.78017558770391864969253425910, −9.913581809165270587081263277278, −9.696913352073337970485985947953, −8.995443944102016733337738042054, −8.477731642746914760838105161970, −7.953886939186400861374322327829, −7.83739345704156595596567940652, −6.91830023491454458142582647377, −6.55915651749569497690275170985, −5.46678604661005193278043078810, −5.16602148805360081393741474967, −4.01692059473226754769728318490, −3.96100024896702990420328380405, −2.83482354145811447731979993100, −2.47860611725090343628991629730, −1.28716060622179546330964871432, −0.913163704047672961162637647557,
0.913163704047672961162637647557, 1.28716060622179546330964871432, 2.47860611725090343628991629730, 2.83482354145811447731979993100, 3.96100024896702990420328380405, 4.01692059473226754769728318490, 5.16602148805360081393741474967, 5.46678604661005193278043078810, 6.55915651749569497690275170985, 6.91830023491454458142582647377, 7.83739345704156595596567940652, 7.953886939186400861374322327829, 8.477731642746914760838105161970, 8.995443944102016733337738042054, 9.696913352073337970485985947953, 9.913581809165270587081263277278, 10.78017558770391864969253425910, 10.93195591872115091424482044338, 11.92387253696974456396438528333, 11.98547328557299080220137017791
