| L(s) = 1 | + (1.05 + 1.37i)3-s + (−1.80 + 1.51i)5-s + (−3.12 + 1.13i)7-s + (−0.773 + 2.89i)9-s + (5.01 + 4.20i)11-s + (−0.627 − 3.56i)13-s + (−3.99 − 0.883i)15-s + (−0.719 − 1.24i)17-s + (2.42 − 4.20i)19-s + (−4.85 − 3.09i)21-s + (5.79 + 2.10i)23-s + (0.100 − 0.568i)25-s + (−4.79 + 1.99i)27-s + (−0.256 + 1.45i)29-s + (7.89 + 2.87i)31-s + ⋯ |
| L(s) = 1 | + (0.609 + 0.793i)3-s + (−0.809 + 0.678i)5-s + (−1.18 + 0.429i)7-s + (−0.257 + 0.966i)9-s + (1.51 + 1.26i)11-s + (−0.174 − 0.987i)13-s + (−1.03 − 0.228i)15-s + (−0.174 − 0.302i)17-s + (0.556 − 0.964i)19-s + (−1.06 − 0.674i)21-s + (1.20 + 0.439i)23-s + (0.0200 − 0.113i)25-s + (−0.923 + 0.384i)27-s + (−0.0477 + 0.270i)29-s + (1.41 + 0.516i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.753451 + 0.886285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.753451 + 0.886285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.05 - 1.37i)T \) |
| good | 5 | \( 1 + (1.80 - 1.51i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (3.12 - 1.13i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-5.01 - 4.20i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.627 + 3.56i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.719 + 1.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.42 + 4.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.79 - 2.10i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.256 - 1.45i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-7.89 - 2.87i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (3.22 + 5.58i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.539 + 3.05i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.55 + 1.30i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.20 - 0.801i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 + (4.52 - 3.79i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.85 + 1.03i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.08 + 6.15i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.303 + 0.525i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.34 + 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.88 - 10.6i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.963 + 5.46i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.09 - 5.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.43 - 6.23i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.49960113743301262678714193094, −11.59498574126827044042138348598, −10.51577360562565629254959060121, −9.535281514679626284476316888413, −8.970047167371133158756795964861, −7.46240222516163991792745395726, −6.71028329036335188615967687825, −4.99901574688908501294566499958, −3.68331793524964693081813405160, −2.85149199295987675217617473828,
1.01020435674161528580336008421, 3.26291746526856382413207768429, 4.14022937945445811575818513041, 6.25573603366468219406433736676, 6.86521087704441028263276312698, 8.213061461683476857371722760434, 8.898594465598197986465595132675, 9.816125150383964440898530044664, 11.55419955109276436565056113616, 12.02030771770925531747535439837
