| L(s) = 1 | + 2·5-s − 18·9-s − 24·13-s + 2·25-s − 80·29-s − 94·37-s + 62·41-s − 36·45-s − 180·53-s + 44·61-s − 48·65-s − 14·73-s + 243·81-s + 238·89-s − 14·97-s − 302·109-s + 448·113-s + 432·117-s + 50·125-s + 127-s + 131-s + 137-s + 139-s − 160·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2/5·5-s − 2·9-s − 1.84·13-s + 2/25·25-s − 2.75·29-s − 2.54·37-s + 1.51·41-s − 4/5·45-s − 3.39·53-s + 0.721·61-s − 0.738·65-s − 0.191·73-s + 3·81-s + 2.67·89-s − 0.144·97-s − 2.77·109-s + 3.96·113-s + 3.69·117-s + 2/5·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.10·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4467295629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4467295629\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 24 T + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )( 1 + 18 T + p^{2} T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 160 T + p^{2} T^{2} )( 1 - 78 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \) |
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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
−11.29905166719647095589353030257, −10.75173039444051713424636902245, −10.48614050325251223733904924575, −9.569656809423703145810550706156, −9.509207406736478001463496967803, −9.054678408906239659996246015826, −8.582645852115927996849715601487, −7.78175845676634739206104981931, −7.76871600921853883001956753435, −7.02431749490582890796766831814, −6.52194547920162278737915447252, −5.74970147081745967007595589089, −5.60200172361185307304694701074, −5.04099318443943936845259117297, −4.52508187023548470730925339140, −3.42722189233929092981039472361, −3.21017195142853361201540508793, −2.27040762533274432061971840140, −1.92523458226620130090730682705, −0.26482182830635757392643846210,
0.26482182830635757392643846210, 1.92523458226620130090730682705, 2.27040762533274432061971840140, 3.21017195142853361201540508793, 3.42722189233929092981039472361, 4.52508187023548470730925339140, 5.04099318443943936845259117297, 5.60200172361185307304694701074, 5.74970147081745967007595589089, 6.52194547920162278737915447252, 7.02431749490582890796766831814, 7.76871600921853883001956753435, 7.78175845676634739206104981931, 8.582645852115927996849715601487, 9.054678408906239659996246015826, 9.509207406736478001463496967803, 9.569656809423703145810550706156, 10.48614050325251223733904924575, 10.75173039444051713424636902245, 11.29905166719647095589353030257
