| L(s) = 1 | + (0.126 − 1.40i)2-s + (−0.698 + 0.843i)3-s + (−1.96 − 0.356i)4-s + (2.03 + 0.937i)5-s + (1.10 + 1.09i)6-s + (1.23 + 0.401i)7-s + (−0.750 + 2.72i)8-s + (0.337 + 1.76i)9-s + (1.57 − 2.74i)10-s + (−0.219 − 1.73i)11-s + (1.67 − 1.41i)12-s + (0.0419 + 0.220i)13-s + (0.721 − 1.69i)14-s + (−2.20 + 1.05i)15-s + (3.74 + 1.40i)16-s + (−0.945 + 3.68i)17-s + ⋯ |
| L(s) = 1 | + (0.0893 − 0.995i)2-s + (−0.403 + 0.487i)3-s + (−0.984 − 0.178i)4-s + (0.907 + 0.419i)5-s + (0.449 + 0.444i)6-s + (0.467 + 0.151i)7-s + (−0.265 + 0.964i)8-s + (0.112 + 0.589i)9-s + (0.498 − 0.866i)10-s + (−0.0660 − 0.522i)11-s + (0.483 − 0.407i)12-s + (0.0116 + 0.0610i)13-s + (0.192 − 0.451i)14-s + (−0.570 + 0.273i)15-s + (0.936 + 0.350i)16-s + (−0.229 + 0.893i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20258 + 0.497275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20258 + 0.497275i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.126 + 1.40i)T \) |
| 5 | \( 1 + (-2.03 - 0.937i)T \) |
| good | 3 | \( 1 + (0.698 - 0.843i)T + (-0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-1.23 - 0.401i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.219 + 1.73i)T + (-10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-0.0419 - 0.220i)T + (-12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (0.945 - 3.68i)T + (-14.8 - 8.18i)T^{2} \) |
| 19 | \( 1 + (4.99 - 4.13i)T + (3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (1.85 + 1.97i)T + (-1.44 + 22.9i)T^{2} \) |
| 29 | \( 1 + (6.22 + 0.391i)T + (28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (-4.02 - 1.03i)T + (27.1 + 14.9i)T^{2} \) |
| 37 | \( 1 + (3.21 - 1.76i)T + (19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (-7.66 - 7.19i)T + (2.57 + 40.9i)T^{2} \) |
| 43 | \( 1 + (-3.84 - 2.79i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (1.13 - 2.86i)T + (-34.2 - 32.1i)T^{2} \) |
| 53 | \( 1 + (-3.38 - 5.33i)T + (-22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (0.877 + 0.413i)T + (37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (-5.80 - 6.18i)T + (-3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.0875 - 1.39i)T + (-66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (-8.36 - 3.31i)T + (51.7 + 48.6i)T^{2} \) |
| 73 | \( 1 + (-3.45 + 1.62i)T + (46.5 - 56.2i)T^{2} \) |
| 79 | \( 1 + (-8.56 + 10.3i)T + (-14.8 - 77.6i)T^{2} \) |
| 83 | \( 1 + (3.77 + 4.56i)T + (-15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (-5.89 - 12.5i)T + (-56.7 + 68.5i)T^{2} \) |
| 97 | \( 1 + (1.94 + 0.122i)T + (96.2 + 12.1i)T^{2} \) |
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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
−10.28728899100580203887939445013, −9.601989645959676326074783225423, −8.590582900870404760438343870702, −7.897150295563840233799261480549, −6.24429920405756944738496391441, −5.68052229948389265106742632518, −4.70067399844958836554148888482, −3.84585306727475836435809221601, −2.49090698225065325819893569292, −1.64192919615597074443908972873,
0.61936943933667268905705157984, 2.14268129458715080427407005831, 3.94675866495104201993878078818, 4.92979393288255311905752083087, 5.63801217288106250016136044896, 6.55502581484478184985992666205, 7.10558118812971810455238698150, 8.064976397106634746415636125217, 9.136050467723778623787462442121, 9.460782923402346874312914184306
