| L(s) = 1 | + (0.891 − 0.453i)2-s + (−2.79 − 0.443i)3-s + (0.587 − 0.809i)4-s + (0.537 − 2.17i)5-s + (−2.69 + 0.875i)6-s + (−0.516 − 3.26i)7-s + (0.156 − 0.987i)8-s + (4.78 + 1.55i)9-s + (−0.506 − 2.17i)10-s + (−0.266 + 3.30i)11-s + (−2.00 + 2.00i)12-s + (1.27 + 2.49i)13-s + (−1.94 − 2.67i)14-s + (−2.46 + 5.83i)15-s + (−0.309 − 0.951i)16-s + (1.30 − 2.57i)17-s + ⋯ |
| L(s) = 1 | + (0.630 − 0.321i)2-s + (−1.61 − 0.255i)3-s + (0.293 − 0.404i)4-s + (0.240 − 0.970i)5-s + (−1.10 + 0.357i)6-s + (−0.195 − 1.23i)7-s + (0.0553 − 0.349i)8-s + (1.59 + 0.518i)9-s + (−0.160 − 0.688i)10-s + (−0.0803 + 0.996i)11-s + (−0.578 + 0.578i)12-s + (0.353 + 0.692i)13-s + (−0.518 − 0.713i)14-s + (−0.636 + 1.50i)15-s + (−0.0772 − 0.237i)16-s + (0.317 − 0.623i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.611526 - 0.684957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.611526 - 0.684957i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.891 + 0.453i)T \) |
| 5 | \( 1 + (-0.537 + 2.17i)T \) |
| 11 | \( 1 + (0.266 - 3.30i)T \) |
| good | 3 | \( 1 + (2.79 + 0.443i)T + (2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (0.516 + 3.26i)T + (-6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 2.49i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.30 + 2.57i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.213 + 0.155i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.16 - 3.16i)T + 23iT^{2} \) |
| 29 | \( 1 + (-6.91 - 5.02i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.80 + 8.63i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.58 - 0.251i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.34 + 3.22i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (0.539 - 0.539i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.09 - 6.91i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (6.64 - 3.38i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (1.61 - 2.22i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (2.35 - 0.763i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (1.62 - 1.62i)T - 67iT^{2} \) |
| 71 | \( 1 + (-3.37 - 10.3i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.88 + 0.297i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.10 + 12.6i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.743 + 0.378i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 17.3iT - 89T^{2} \) |
| 97 | \( 1 + (2.33 + 4.57i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−13.14430551720955606920930761353, −12.32097836515382033090433090899, −11.53572147463081757816288270999, −10.50379695170510954259127650460, −9.554784993275093764145545758576, −7.38723617686319119297061338136, −6.40309804332339807830730484060, −5.10082052838870085055181116350, −4.32355208377456433142948494265, −1.17400480006738473104269577991,
3.13846200300891856234456330123, 5.10491505974717030337114956503, 6.02315315187493854326153796300, 6.56124369362145959986153099591, 8.393286416726838327625383014229, 10.19605062056217899020181828254, 10.98253269277182031237429436488, 11.87539283100757403761888690861, 12.67919987356466229547369787197, 13.95877313214182190642651037519
