| L(s) = 1 | + (−1.37 + 0.339i)2-s + (0.195 − 0.980i)3-s + (1.76 − 0.931i)4-s + (2.28 + 1.52i)5-s + (0.0649 + 1.41i)6-s + (0.137 − 0.0568i)7-s + (−2.11 + 1.87i)8-s + (−0.923 − 0.382i)9-s + (−3.65 − 1.32i)10-s + (−1.06 + 0.211i)11-s + (−0.568 − 1.91i)12-s + (4.19 − 2.80i)13-s + (−0.169 + 0.124i)14-s + (1.94 − 1.94i)15-s + (2.26 − 3.29i)16-s + (3.99 + 3.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.970 + 0.239i)2-s + (0.112 − 0.566i)3-s + (0.884 − 0.465i)4-s + (1.02 + 0.682i)5-s + (0.0265 + 0.576i)6-s + (0.0518 − 0.0214i)7-s + (−0.747 + 0.664i)8-s + (−0.307 − 0.127i)9-s + (−1.15 − 0.417i)10-s + (−0.320 + 0.0636i)11-s + (−0.164 − 0.553i)12-s + (1.16 − 0.776i)13-s + (−0.0452 + 0.0333i)14-s + (0.501 − 0.501i)15-s + (0.565 − 0.824i)16-s + (0.968 + 0.968i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.967316 - 0.00913333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.967316 - 0.00913333i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.339i)T \) |
| 3 | \( 1 + (-0.195 + 0.980i)T \) |
| good | 5 | \( 1 + (-2.28 - 1.52i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.137 + 0.0568i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (1.06 - 0.211i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-4.19 + 2.80i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-3.99 - 3.99i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.247 - 0.370i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.503 + 1.21i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (3.33 + 0.663i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 7.31iT - 31T^{2} \) |
| 37 | \( 1 + (4.18 - 6.26i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-3.09 + 7.47i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.61 - 8.09i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (3.27 + 3.27i)T + 47iT^{2} \) |
| 53 | \( 1 + (10.2 - 2.04i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (10.8 + 7.24i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (1.01 - 5.11i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (2.90 - 14.5i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (8.79 - 3.64i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (1.22 + 0.505i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-8.40 + 8.40i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.437 - 0.655i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-3.63 - 8.76i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 1.03iT - 97T^{2} \) |
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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
−12.56080183170651204031361081029, −11.19667960853587701534511287867, −10.44676052166195723483734747988, −9.622352111049969345489445203152, −8.378640743966332598259300159735, −7.58860624136225184005659575554, −6.25917160523194140042218196239, −5.78856360378824859761353638001, −3.03980086316046161192074299377, −1.59757956651186585383272411029,
1.60360816088254664456730124071, 3.32930067939382510770518892826, 5.14681133572551975885086060712, 6.30413983130185231623722002471, 7.73809547080733580076269733857, 8.999876857079791274127187542362, 9.343349876192018697489447021472, 10.40474980528628448555940888082, 11.28331859491490194822552469641, 12.37354968427787011022993315005
