| L(s) = 1 | + (−0.311 + 1.37i)2-s + (1.65 + 0.523i)3-s + (−1.80 − 0.860i)4-s + (−0.573 + 2.16i)5-s + (−1.23 + 2.11i)6-s + 1.39i·7-s + (1.74 − 2.22i)8-s + (2.45 + 1.72i)9-s + (−2.80 − 1.46i)10-s + (0.762 + 0.554i)11-s + (−2.53 − 2.36i)12-s + (−0.565 + 0.410i)13-s + (−1.92 − 0.436i)14-s + (−2.07 + 3.26i)15-s + (2.52 + 3.10i)16-s + (−7.22 − 2.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.220 + 0.975i)2-s + (0.953 + 0.302i)3-s + (−0.902 − 0.430i)4-s + (−0.256 + 0.966i)5-s + (−0.504 + 0.863i)6-s + 0.528i·7-s + (0.618 − 0.785i)8-s + (0.817 + 0.576i)9-s + (−0.886 − 0.463i)10-s + (0.229 + 0.167i)11-s + (−0.730 − 0.682i)12-s + (−0.156 + 0.113i)13-s + (−0.515 − 0.116i)14-s + (−0.536 + 0.843i)15-s + (0.630 + 0.776i)16-s + (−1.75 − 0.569i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.504484 + 1.24959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.504484 + 1.24959i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.311 - 1.37i)T \) |
| 3 | \( 1 + (-1.65 - 0.523i)T \) |
| 5 | \( 1 + (0.573 - 2.16i)T \) |
| good | 7 | \( 1 - 1.39iT - 7T^{2} \) |
| 11 | \( 1 + (-0.762 - 0.554i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.565 - 0.410i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (7.22 + 2.34i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.528 - 0.171i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.42 - 2.49i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.38 + 1.10i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.35 - 3.03i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.815 - 0.592i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.88 + 6.71i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 + (-1.76 - 5.42i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.78 + 1.55i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.62 + 4.08i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.0279 - 0.0203i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.99 - 1.62i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.97 + 9.16i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.64 - 5.55i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-8.04 + 2.61i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.685i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (4.97 - 6.84i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.529 + 1.62i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.15850680327940656459699295867, −10.86242028697338864514402069552, −9.958417746913718625456461593242, −9.036830493098405316943543733678, −8.350545049549110574443324118497, −7.17594177753748717755743321511, −6.62413505692273891483346284071, −5.03882818532975005183696109393, −3.91164065017115941600940798542, −2.47491497623074756661084247913,
1.07236573038573634286561310069, 2.56415449516764408543829998246, 3.99102775548203411461953192843, 4.70556043251269997449803056484, 6.71552732554126971381299957532, 8.057026948704701668349134026772, 8.600130946501371577506411961618, 9.411135964631089795346765903666, 10.36850684819448777765740172847, 11.48497759684114216833926675580
