| L(s) = 1 | + (−1.38 − 0.282i)2-s + (−0.980 + 0.195i)3-s + (1.84 + 0.782i)4-s + (2.00 + 2.99i)5-s + (1.41 + 0.00649i)6-s + (−3.93 − 1.63i)7-s + (−2.32 − 1.60i)8-s + (0.923 − 0.382i)9-s + (−1.92 − 4.71i)10-s + (−0.787 + 3.95i)11-s + (−1.95 − 0.408i)12-s + (−0.507 + 0.760i)13-s + (4.99 + 3.37i)14-s + (−2.54 − 2.54i)15-s + (2.77 + 2.87i)16-s + (−3.70 + 3.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.979 − 0.199i)2-s + (−0.566 + 0.112i)3-s + (0.920 + 0.391i)4-s + (0.894 + 1.33i)5-s + (0.577 + 0.00265i)6-s + (−1.48 − 0.616i)7-s + (−0.823 − 0.566i)8-s + (0.307 − 0.127i)9-s + (−0.609 − 1.49i)10-s + (−0.237 + 1.19i)11-s + (−0.565 − 0.117i)12-s + (−0.140 + 0.210i)13-s + (1.33 + 0.901i)14-s + (−0.657 − 0.657i)15-s + (0.693 + 0.719i)16-s + (−0.898 + 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.307922 + 0.403448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.307922 + 0.403448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.282i)T \) |
| 3 | \( 1 + (0.980 - 0.195i)T \) |
| good | 5 | \( 1 + (-2.00 - 2.99i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (3.93 + 1.63i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.787 - 3.95i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (0.507 - 0.760i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (3.70 - 3.70i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.20 - 0.804i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-2.69 - 6.51i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.13 + 5.71i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 2.46iT - 31T^{2} \) |
| 37 | \( 1 + (3.29 - 2.20i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-0.457 - 1.10i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-6.68 - 1.33i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-2.46 + 2.46i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.69 + 8.52i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-3.49 - 5.22i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-8.64 + 1.71i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.326 + 0.0648i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (10.0 + 4.15i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-12.5 + 5.19i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (3.74 + 3.74i)T + 79iT^{2} \) |
| 83 | \( 1 + (-4.11 - 2.74i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (2.28 - 5.52i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 0.186iT - 97T^{2} \) |
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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
−12.76696933417999177715743069463, −11.49665933359434724539696562831, −10.52081296477247294422001543484, −9.955061336739918255186661192382, −9.406312440207513015010770828012, −7.39375828960540874316219677770, −6.76631694082174023722269972992, −5.99833585645433723226275186245, −3.64763372088160800789284354704, −2.23055759210039040232680335523,
0.62354203954274323677010397353, 2.67242224280307850855826303891, 5.23629664897998143706954906308, 5.99235863539111748844276651256, 6.94106052580914322353127081001, 8.754269080161292016335345564159, 9.038467691584805703111454876106, 10.05987149765914402324067822533, 11.08587062683982581716945410311, 12.38033862253035488645378828508
