| L(s) = 1 | + 76·5-s + 308·13-s − 228·17-s + 3.08e3·25-s + 140·29-s − 1.54e3·37-s − 1.54e3·41-s + 4.73e3·49-s + 8.46e3·53-s + 1.14e4·61-s + 2.34e4·65-s + 1.95e3·73-s − 1.73e4·85-s − 1.15e4·89-s + 2.14e4·97-s − 3.23e4·101-s + 3.14e4·109-s + 1.01e4·113-s − 1.56e4·121-s + 7.69e4·125-s + 127-s + 131-s + 137-s + 139-s + 1.06e4·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3.03·5-s + 1.82·13-s − 0.788·17-s + 4.93·25-s + 0.166·29-s − 1.13·37-s − 0.916·41-s + 1.97·49-s + 3.01·53-s + 3.07·61-s + 5.54·65-s + 0.367·73-s − 2.39·85-s − 1.45·89-s + 2.28·97-s − 3.17·101-s + 2.64·109-s + 0.791·113-s − 1.06·121-s + 4.92·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 0.506·145-s + 4.50e−5·149-s + 4.38e−5·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(6.952441868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.952441868\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 38 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 4738 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15662 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 154 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 114 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 578 p^{2} T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 551938 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 70 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1207042 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 774 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 770 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1173202 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 7043458 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4230 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 22606546 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5722 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 27371026 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 242818 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 978 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 178306 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 87841042 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5778 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10738 T + p^{4} T^{2} )^{2} \) |
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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
−11.15251059745530964615322886364, −10.86207994306704604417042956008, −10.24092626683032328603276367207, −10.04746690216280609819299070061, −9.686542157983057279563647170873, −8.890187937960841277263628812992, −8.644024930681722023773157087092, −8.562299237031459295063095344255, −7.22664470300831959913613977192, −6.84776746734996903323998408684, −6.29507348320230371554782774279, −5.96956685894838776852951455461, −5.33848882105144063271875090626, −5.28372496349306524784742203049, −4.12093549917944988029783670264, −3.52771295680318269968040255736, −2.37096116386355390464101772103, −2.27006394364569775279589314296, −1.43954060328978655020724641164, −0.878189081937637644191925316242,
0.878189081937637644191925316242, 1.43954060328978655020724641164, 2.27006394364569775279589314296, 2.37096116386355390464101772103, 3.52771295680318269968040255736, 4.12093549917944988029783670264, 5.28372496349306524784742203049, 5.33848882105144063271875090626, 5.96956685894838776852951455461, 6.29507348320230371554782774279, 6.84776746734996903323998408684, 7.22664470300831959913613977192, 8.562299237031459295063095344255, 8.644024930681722023773157087092, 8.890187937960841277263628812992, 9.686542157983057279563647170873, 10.04746690216280609819299070061, 10.24092626683032328603276367207, 10.86207994306704604417042956008, 11.15251059745530964615322886364
