| L(s) = 1 | + 18·3-s − 64·5-s − 98·7-s + 243·9-s − 540·11-s + 204·13-s − 1.15e3·15-s + 304·17-s + 1.12e3·19-s − 1.76e3·21-s + 940·23-s − 2.18e3·25-s + 2.91e3·27-s + 932·29-s + 1.64e4·31-s − 9.72e3·33-s + 6.27e3·35-s − 1.76e3·37-s + 3.67e3·39-s − 2.55e3·41-s + 2.46e4·43-s − 1.55e4·45-s + 3.67e4·47-s + 7.20e3·49-s + 5.47e3·51-s + 9.16e3·53-s + 3.45e4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.14·5-s − 0.755·7-s + 9-s − 1.34·11-s + 0.334·13-s − 1.32·15-s + 0.255·17-s + 0.711·19-s − 0.872·21-s + 0.370·23-s − 0.698·25-s + 0.769·27-s + 0.205·29-s + 3.06·31-s − 1.55·33-s + 0.865·35-s − 0.211·37-s + 0.386·39-s − 0.237·41-s + 2.03·43-s − 1.14·45-s + 2.42·47-s + 3/7·49-s + 0.294·51-s + 0.448·53-s + 1.54·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.864606427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.864606427\) |
| \(L(\frac{7}{2})\) |
|
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 | $D_{4}$ | \( 1 + 64 T + 6278 T^{2} + 64 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 540 T + 170902 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 204 T + 498014 T^{2} - 204 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 304 T + 272222 T^{2} - 304 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1120 T - 188298 T^{2} - 1120 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 940 T + 8072750 T^{2} - 940 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 932 T + 41139854 T^{2} - 932 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 16408 T + 122456382 T^{2} - 16408 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1764 T + 117649454 T^{2} + 1764 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2552 T - 47492578 T^{2} + 2552 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 24632 T + 444919878 T^{2} - 24632 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 36760 T + 728144990 T^{2} - 36760 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9164 T + 526936814 T^{2} - 9164 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 39888 T + 1364348230 T^{2} - 39888 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 25084 T + 1327295502 T^{2} + 25084 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2592 T + 466901846 T^{2} - 2592 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 35508 T + 2992314574 T^{2} - 35508 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17188 T + 3868575366 T^{2} + 17188 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 95800 T + 6912129054 T^{2} - 95800 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 65352 T + 5470335958 T^{2} + 65352 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 23000 T + 9266496062 T^{2} + 23000 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 108388 T + 14152685814 T^{2} + 108388 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.76369386288637598233115183051, −10.48108282958140639165345696746, −9.796871795521251433747920260483, −9.792923732357535811659437980389, −8.877903386482260819738267938439, −8.643817319318985614139412850841, −7.921472774005890649044041785065, −7.902888211080406742560822979675, −7.22335309741655018471536186417, −6.97915552497176790829687037549, −5.90921688628840550378988581801, −5.80387098553843759783705087714, −4.58345863832954192183181489964, −4.52589599392285324292687284254, −3.47167453273810960455547794605, −3.46449914107610976806436296690, −2.50478223405585195365999899732, −2.35678817938169513331655673666, −0.873912671042865415000643638330, −0.63614273554842050961409129086,
0.63614273554842050961409129086, 0.873912671042865415000643638330, 2.35678817938169513331655673666, 2.50478223405585195365999899732, 3.46449914107610976806436296690, 3.47167453273810960455547794605, 4.52589599392285324292687284254, 4.58345863832954192183181489964, 5.80387098553843759783705087714, 5.90921688628840550378988581801, 6.97915552497176790829687037549, 7.22335309741655018471536186417, 7.902888211080406742560822979675, 7.921472774005890649044041785065, 8.643817319318985614139412850841, 8.877903386482260819738267938439, 9.792923732357535811659437980389, 9.796871795521251433747920260483, 10.48108282958140639165345696746, 10.76369386288637598233115183051
