| L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.485 + 2.75i)3-s + (0.766 − 0.642i)4-s + (−1.79 − 1.50i)5-s + (0.485 + 2.75i)6-s + (0.642 + 1.11i)7-s + (0.500 − 0.866i)8-s + (−4.51 − 1.64i)9-s + (−2.20 − 0.801i)10-s + (−2.87 + 4.98i)11-s + (1.39 + 2.41i)12-s + (−0.0528 − 0.299i)13-s + (0.983 + 0.825i)14-s + (5.01 − 4.21i)15-s + (0.173 − 0.984i)16-s + (−3.93 + 1.43i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.280 + 1.58i)3-s + (0.383 − 0.321i)4-s + (−0.803 − 0.673i)5-s + (0.198 + 1.12i)6-s + (0.242 + 0.420i)7-s + (0.176 − 0.306i)8-s + (−1.50 − 0.547i)9-s + (−0.696 − 0.253i)10-s + (−0.867 + 1.50i)11-s + (0.403 + 0.698i)12-s + (−0.0146 − 0.0831i)13-s + (0.262 + 0.220i)14-s + (1.29 − 1.08i)15-s + (0.0434 − 0.246i)16-s + (−0.954 + 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.203008 + 0.995149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203008 + 0.995149i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (0.485 - 2.75i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (1.79 + 1.50i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.642 - 1.11i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.87 - 4.98i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0528 + 0.299i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.93 - 1.43i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (4.96 - 4.16i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.93 + 1.06i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.22 - 5.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 + (-0.871 + 4.94i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.757 + 0.635i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.12 - 1.50i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.52 - 2.11i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.11 + 1.13i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.39 - 7.04i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.11 + 1.49i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.38 - 2.83i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.393 - 2.23i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.48 - 8.44i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.88 - 8.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.61 - 14.8i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-16.3 + 5.93i)T + (74.3 - 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.73268067177427114392772596334, −10.09459618811043248073312769191, −9.294626569272776282918124909323, −8.351878907297399862365146019791, −7.33849245683104016020603354194, −5.86235508321739628109486619423, −4.97005914485633517991489298249, −4.46598191228660652415866360542, −3.73951643067628138645731200114, −2.26755704572043958685791265507,
0.41056522175886852397030327106, 2.26492844935786865596362085726, 3.27173445200985380947209665525, 4.55348213803709891620112498791, 5.96692018810308164176107230121, 6.38968096307143657760838912682, 7.54797695559602259270527103057, 7.74279429222354496795149710973, 8.690501424422472445516845355437, 10.50589088763792038676862579920
