| L(s) = 1 | + (−0.338 − 1.37i)2-s + (1.70 + 0.314i)3-s + (−1.77 + 0.930i)4-s + (0.479 + 0.415i)5-s + (−0.145 − 2.44i)6-s + (1.80 + 2.81i)7-s + (1.87 + 2.11i)8-s + (2.80 + 1.07i)9-s + (0.407 − 0.798i)10-s + (−0.284 + 1.98i)11-s + (−3.30 + 1.02i)12-s + (−1.90 − 1.22i)13-s + (3.25 − 3.43i)14-s + (0.685 + 0.857i)15-s + (2.26 − 3.29i)16-s + (1.27 − 4.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.239 − 0.970i)2-s + (0.983 + 0.181i)3-s + (−0.885 + 0.465i)4-s + (0.214 + 0.185i)5-s + (−0.0592 − 0.998i)6-s + (0.684 + 1.06i)7-s + (0.663 + 0.747i)8-s + (0.933 + 0.357i)9-s + (0.128 − 0.252i)10-s + (−0.0859 + 0.597i)11-s + (−0.954 + 0.296i)12-s + (−0.527 − 0.338i)13-s + (0.869 − 0.919i)14-s + (0.176 + 0.221i)15-s + (0.566 − 0.823i)16-s + (0.308 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51032 - 0.303150i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51032 - 0.303150i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.338 + 1.37i)T \) |
| 3 | \( 1 + (-1.70 - 0.314i)T \) |
| 23 | \( 1 + (4.70 - 0.934i)T \) |
| good | 5 | \( 1 + (-0.479 - 0.415i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.80 - 2.81i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.284 - 1.98i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.90 + 1.22i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.27 + 4.33i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.200 + 0.683i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.681 - 2.32i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-4.45 + 2.03i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.26 + 6.07i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (6.37 + 5.52i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.30 - 3.33i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 + (-5.97 - 9.29i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (12.5 + 8.06i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (0.377 + 0.827i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (3.50 - 0.504i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.747 - 5.20i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (5.36 - 1.57i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (0.930 - 1.44i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.69 - 8.87i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.443 - 0.202i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-7.32 + 8.45i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−12.04050200313932661689862967897, −10.72340992588189160611871981912, −9.846028491458571438587922069965, −9.139661244316880595072424374264, −8.236974298354669486422641205748, −7.38638470644284870937303852236, −5.36674207773781117200462101242, −4.32736637125935264648954130710, −2.81313332811354175659044654655, −2.02919339542186778614247201610,
1.48616115618383448863324874175, 3.71780635633990187839280777638, 4.72419493161839091207205691763, 6.18095641008417151155177292200, 7.30996830617020428271842493320, 8.027972682846453493848783893479, 8.754475569504775810429418167480, 9.899251165158918118665280196865, 10.55578962963892419512007940201, 12.16669430953488948549246773526
