| 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
−10.50589088763792038676862579920, −8.690501424422472445516845355437, −7.74279429222354496795149710973, −7.54797695559602259270527103057, −6.38968096307143657760838912682, −5.96692018810308164176107230121, −4.55348213803709891620112498791, −3.27173445200985380947209665525, −2.26492844935786865596362085726, −0.41056522175886852397030327106,
2.26755704572043958685791265507, 3.73951643067628138645731200114, 4.46598191228660652415866360542, 4.97005914485633517991489298249, 5.86235508321739628109486619423, 7.33849245683104016020603354194, 8.351878907297399862365146019791, 9.294626569272776282918124909323, 10.09459618811043248073312769191, 10.73268067177427114392772596334
