| L(s) = 1 | + (−0.256 − 1.39i)2-s + (−0.809 + 1.53i)3-s + (−1.86 + 0.714i)4-s + (2.40 + 0.105i)5-s + (2.33 + 0.732i)6-s + (3.60 + 2.52i)7-s + (1.47 + 2.41i)8-s + (−1.68 − 2.47i)9-s + (−0.472 − 3.37i)10-s + (3.60 + 3.29i)11-s + (0.418 − 3.43i)12-s + (−0.393 + 1.24i)13-s + (2.58 − 5.66i)14-s + (−2.10 + 3.59i)15-s + (2.97 − 2.66i)16-s + (−2.98 + 5.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.181 − 0.983i)2-s + (−0.467 + 0.884i)3-s + (−0.933 + 0.357i)4-s + (1.07 + 0.0469i)5-s + (0.954 + 0.299i)6-s + (1.36 + 0.954i)7-s + (0.521 + 0.853i)8-s + (−0.562 − 0.826i)9-s + (−0.149 − 1.06i)10-s + (1.08 + 0.994i)11-s + (0.120 − 0.992i)12-s + (−0.109 + 0.345i)13-s + (0.690 − 1.51i)14-s + (−0.544 + 0.929i)15-s + (0.744 − 0.667i)16-s + (−0.723 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36849 + 0.543419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36849 + 0.543419i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.256 + 1.39i)T \) |
| 3 | \( 1 + (0.809 - 1.53i)T \) |
| good | 5 | \( 1 + (-2.40 - 0.105i)T + (4.98 + 0.435i)T^{2} \) |
| 7 | \( 1 + (-3.60 - 2.52i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-3.60 - 3.29i)T + (0.958 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.393 - 1.24i)T + (-10.6 - 7.45i)T^{2} \) |
| 17 | \( 1 + (2.98 - 5.16i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.52 + 2.70i)T + (4.91 + 18.3i)T^{2} \) |
| 23 | \( 1 + (3.38 + 4.83i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (5.10 + 2.65i)T + (16.6 + 23.7i)T^{2} \) |
| 31 | \( 1 + (-7.05 - 1.24i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.665 - 0.867i)T + (-9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-0.766 - 8.76i)T + (-40.3 + 7.11i)T^{2} \) |
| 43 | \( 1 + (-2.29 + 2.50i)T + (-3.74 - 42.8i)T^{2} \) |
| 47 | \( 1 + (0.994 - 0.175i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.87 + 9.36i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (0.570 - 13.0i)T + (-58.7 - 5.14i)T^{2} \) |
| 61 | \( 1 + (-1.71 - 2.68i)T + (-25.7 + 55.2i)T^{2} \) |
| 67 | \( 1 + (-0.963 + 3.05i)T + (-54.8 - 38.4i)T^{2} \) |
| 71 | \( 1 + (-4.89 + 1.31i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.79 - 1.28i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.87 + 4.08i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (11.8 + 6.16i)T + (47.6 + 67.9i)T^{2} \) |
| 89 | \( 1 + (10.2 + 2.74i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-4.29 - 1.56i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.24479463408309616278421268602, −9.603956335051403718580307303379, −8.846562971198290519281679163898, −8.304809488898101043839506188686, −6.51169200100540703736402741044, −5.68970620687305329458587712526, −4.55878665041672791907115779273, −4.25110075596682064051567364664, −2.38246276665891545151815985606, −1.71729471362409000827751942823,
0.873503129460475054667797024995, 1.89176505207673827221387886552, 4.02939808271156735463088653685, 5.13598791610610044605385094139, 5.83607244980832749302894682184, 6.60716535933017633323099790806, 7.44305924495408303090309209281, 8.158468998060118520802232990710, 8.989023318710077775116138984712, 9.955675151243842987797849665117
