| L(s) = 1 | + (−1.54 − 0.785i)3-s + (1.22 − 1.03i)5-s + (−0.759 + 0.276i)7-s + (1.76 + 2.42i)9-s + (−2.65 − 2.23i)11-s + (−0.365 − 2.07i)13-s + (−2.70 + 0.626i)15-s + (−1.10 − 1.90i)17-s + (−0.0363 + 0.0629i)19-s + (1.38 + 0.169i)21-s + (0.552 + 0.201i)23-s + (−0.421 + 2.39i)25-s + (−0.821 − 5.13i)27-s + (0.154 − 0.874i)29-s + (−3.97 − 1.44i)31-s + ⋯ |
| L(s) = 1 | + (−0.891 − 0.453i)3-s + (0.549 − 0.460i)5-s + (−0.287 + 0.104i)7-s + (0.588 + 0.808i)9-s + (−0.801 − 0.672i)11-s + (−0.101 − 0.574i)13-s + (−0.698 + 0.161i)15-s + (−0.267 − 0.462i)17-s + (−0.00834 + 0.0144i)19-s + (0.303 + 0.0370i)21-s + (0.115 + 0.0419i)23-s + (−0.0843 + 0.478i)25-s + (−0.158 − 0.987i)27-s + (0.0286 − 0.162i)29-s + (−0.714 − 0.259i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0740981 - 0.535395i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0740981 - 0.535395i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.54 + 0.785i)T \) |
| good | 5 | \( 1 + (-1.22 + 1.03i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.759 - 0.276i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.65 + 2.23i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.365 + 2.07i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.10 + 1.90i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0363 - 0.0629i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.552 - 0.201i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.154 + 0.874i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.97 + 1.44i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.82 + 3.16i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.43 + 8.13i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.703 + 0.589i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (10.8 - 3.96i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 + (-0.409 + 0.343i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (10.8 - 3.94i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.11 + 12.0i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 - 2.67i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.82 - 6.62i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.14 - 12.1i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.69 + 15.2i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.93 + 13.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.45 + 2.06i)T + (16.8 + 95.5i)T^{2} \) |
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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
−9.875454277495662335869744568063, −9.001084453240914869986739985513, −7.958149140257637403009403385089, −7.18043559080206224633333760348, −6.07815186962486382114615915418, −5.50723690267432358442316060282, −4.72706354132834426528530526455, −3.14501701270898762574337255428, −1.79565166083360318514789234523, −0.27967526872909581548879537650,
1.82398922762580333097853229580, 3.23411626818790712716503271592, 4.48318041261673372969287234059, 5.22879158100471878908113180073, 6.34206849425110811391789306951, 6.78034028223484604302143767060, 7.924103545646873817150380763058, 9.162138147730752349261370326269, 9.959305718168091587328522923956, 10.41105702354595323678314322031
