| L(s) = 1 | + (1.16 + 0.801i)2-s + (−0.695 − 1.58i)3-s + (0.715 + 1.86i)4-s + (0.743 + 1.39i)5-s + (0.461 − 2.40i)6-s + (2.39 − 0.476i)7-s + (−0.663 + 2.74i)8-s + (−2.03 + 2.20i)9-s + (−0.248 + 2.21i)10-s + (1.22 − 1.49i)11-s + (2.46 − 2.43i)12-s + (−0.813 + 1.52i)13-s + (3.17 + 1.36i)14-s + (1.68 − 2.14i)15-s + (−2.97 + 2.67i)16-s + (1.98 + 0.822i)17-s + ⋯ |
| L(s) = 1 | + (0.823 + 0.566i)2-s + (−0.401 − 0.915i)3-s + (0.357 + 0.933i)4-s + (0.332 + 0.621i)5-s + (0.188 − 0.982i)6-s + (0.905 − 0.180i)7-s + (−0.234 + 0.972i)8-s + (−0.677 + 0.735i)9-s + (−0.0785 + 0.700i)10-s + (0.370 − 0.450i)11-s + (0.711 − 0.702i)12-s + (−0.225 + 0.422i)13-s + (0.848 + 0.364i)14-s + (0.436 − 0.553i)15-s + (−0.744 + 0.668i)16-s + (0.481 + 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00251 + 0.688879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00251 + 0.688879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 - 0.801i)T \) |
| 3 | \( 1 + (0.695 + 1.58i)T \) |
| good | 5 | \( 1 + (-0.743 - 1.39i)T + (-2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (-2.39 + 0.476i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 1.49i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.813 - 1.52i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.98 - 0.822i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-7.57 + 2.29i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (5.64 - 3.77i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (5.74 + 6.99i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-0.0203 - 0.0203i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.53 - 1.07i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (5.60 - 3.74i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (0.787 - 0.0775i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (2.76 + 1.14i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.17 + 1.42i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (5.16 + 9.66i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (13.9 + 1.37i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-1.48 + 15.0i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (0.904 - 0.179i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.60 + 8.07i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.14 - 7.59i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (5.74 - 1.74i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (6.36 - 9.52i)T + (-34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (4.53 - 4.53i)T - 97iT^{2} \) |
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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
−11.58796582761656177160681926881, −11.06752529126513335485297064552, −9.568964515679183466856039461841, −8.041196820949642001653097514435, −7.61377496908489508298364516345, −6.55692179964445508428464708814, −5.79594841959892412098657861669, −4.81264070963355458934704226738, −3.28106197267261773603760851231, −1.86586296868531678822180436391,
1.46046826427244068371855758700, 3.22202248192950623080329703934, 4.42033267709622109032589190329, 5.25042576116831414549298317689, 5.79886323280741972567172935647, 7.37860021921896669432842345891, 8.836641328377239983713121543240, 9.716679356774016432521586462049, 10.38023120384840431394128736390, 11.43202644813562551397454238187
