| 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
−11.48497759684114216833926675580, −10.36850684819448777765740172847, −9.411135964631089795346765903666, −8.600130946501371577506411961618, −8.057026948704701668349134026772, −6.71552732554126971381299957532, −4.70556043251269997449803056484, −3.99102775548203411461953192843, −2.56415449516764408543829998246, −1.07236573038573634286561310069,
2.47491497623074756661084247913, 3.91164065017115941600940798542, 5.03882818532975005183696109393, 6.62413505692273891483346284071, 7.17594177753748717755743321511, 8.350545049549110574443324118497, 9.036830493098405316943543733678, 9.958417746913718625456461593242, 10.86242028697338864514402069552, 12.15850680327940656459699295867
