| L(s) = 1 | + (0.706 + 2.54i)2-s + (−1.23 − 1.21i)3-s + (−4.25 + 2.56i)4-s + (−3.25 + 0.176i)5-s + (2.22 − 3.99i)6-s + (−0.210 − 1.28i)7-s + (−5.69 − 5.39i)8-s + (0.0428 + 2.99i)9-s + (−2.74 − 8.15i)10-s + (1.93 + 3.64i)11-s + (8.36 + 2.01i)12-s + (−1.56 − 0.623i)13-s + (3.11 − 1.43i)14-s + (4.22 + 3.73i)15-s + (5.03 − 9.50i)16-s + (−3.37 − 0.553i)17-s + ⋯ |
| L(s) = 1 | + (0.499 + 1.79i)2-s + (−0.712 − 0.702i)3-s + (−2.12 + 1.28i)4-s + (−1.45 + 0.0788i)5-s + (0.907 − 1.63i)6-s + (−0.0794 − 0.484i)7-s + (−2.01 − 1.90i)8-s + (0.0142 + 0.999i)9-s + (−0.868 − 2.57i)10-s + (0.582 + 1.09i)11-s + (2.41 + 0.582i)12-s + (−0.433 − 0.172i)13-s + (0.831 − 0.384i)14-s + (1.09 + 0.965i)15-s + (1.25 − 2.37i)16-s + (−0.819 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.171021 - 0.346843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171021 - 0.346843i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.23 + 1.21i)T \) |
| 59 | \( 1 + (6.38 - 4.27i)T \) |
| good | 2 | \( 1 + (-0.706 - 2.54i)T + (-1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (3.25 - 0.176i)T + (4.97 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.210 + 1.28i)T + (-6.63 + 2.23i)T^{2} \) |
| 11 | \( 1 + (-1.93 - 3.64i)T + (-6.17 + 9.10i)T^{2} \) |
| 13 | \( 1 + (1.56 + 0.623i)T + (9.43 + 8.94i)T^{2} \) |
| 17 | \( 1 + (3.37 + 0.553i)T + (16.1 + 5.42i)T^{2} \) |
| 19 | \( 1 + (5.39 - 6.35i)T + (-3.07 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.326 - 0.248i)T + (6.15 + 22.1i)T^{2} \) |
| 29 | \( 1 + (-6.81 - 1.89i)T + (24.8 + 14.9i)T^{2} \) |
| 31 | \( 1 + (0.0480 - 0.0408i)T + (5.01 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.21 - 3.39i)T + (-2.00 + 36.9i)T^{2} \) |
| 41 | \( 1 + (2.06 + 2.71i)T + (-10.9 + 39.5i)T^{2} \) |
| 43 | \( 1 + (4.01 + 2.12i)T + (24.1 + 35.5i)T^{2} \) |
| 47 | \( 1 + (0.111 - 2.05i)T + (-46.7 - 5.08i)T^{2} \) |
| 53 | \( 1 + (-1.69 + 5.03i)T + (-42.1 - 32.0i)T^{2} \) |
| 61 | \( 1 + (6.57 - 1.82i)T + (52.2 - 31.4i)T^{2} \) |
| 67 | \( 1 + (7.13 - 7.52i)T + (-3.62 - 66.9i)T^{2} \) |
| 71 | \( 1 + (3.83 + 0.208i)T + (70.5 + 7.67i)T^{2} \) |
| 73 | \( 1 + (-2.50 - 5.41i)T + (-47.2 + 55.6i)T^{2} \) |
| 79 | \( 1 + (0.654 + 0.964i)T + (-29.2 + 73.3i)T^{2} \) |
| 83 | \( 1 + (-9.57 + 2.10i)T + (75.3 - 34.8i)T^{2} \) |
| 89 | \( 1 + (-2.24 + 8.09i)T + (-76.2 - 45.8i)T^{2} \) |
| 97 | \( 1 + (1.16 - 2.51i)T + (-62.7 - 73.9i)T^{2} \) |
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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
−13.40887032586060231199119196875, −12.44076537582525106536188190225, −11.93106617188337174503791098268, −10.37134575700769930249887506868, −8.590484220863068579321224708049, −7.73352979872266949925419101333, −7.05549771257084244381408991140, −6.31930556135765642288103767423, −4.77052415319887422112653529584, −4.06268194752776366445669755011,
0.33117769305239505162992297960, 2.96377387191375079156899905371, 4.14806841904929912725678143360, 4.76224916999602501631896882035, 6.34874058382801293037043591505, 8.597301548819135904480001518718, 9.258921202160129745358042114220, 10.69159425347813238366480782952, 11.18053730263066921325178507318, 11.86667458728930805417784190451
