| L(s) = 1 | + (−1.38 − 0.278i)2-s + (2.24 + 0.882i)3-s + (1.84 + 0.772i)4-s + (−0.933 + 0.0699i)5-s + (−2.87 − 1.85i)6-s + (−0.726 + 1.25i)7-s + (−2.34 − 1.58i)8-s + (2.07 + 1.92i)9-s + (1.31 + 0.162i)10-s + (3.74 + 0.855i)11-s + (3.46 + 3.36i)12-s + (1.53 + 2.24i)13-s + (1.35 − 1.54i)14-s + (−2.16 − 0.666i)15-s + (2.80 + 2.84i)16-s + (0.126 − 1.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.980 − 0.196i)2-s + (1.29 + 0.509i)3-s + (0.922 + 0.386i)4-s + (−0.417 + 0.0312i)5-s + (−1.17 − 0.755i)6-s + (−0.274 + 0.475i)7-s + (−0.828 − 0.560i)8-s + (0.693 + 0.643i)9-s + (0.415 + 0.0515i)10-s + (1.13 + 0.258i)11-s + (1.00 + 0.971i)12-s + (0.425 + 0.623i)13-s + (0.362 − 0.412i)14-s + (−0.557 − 0.172i)15-s + (0.701 + 0.712i)16-s + (0.0306 − 0.408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16621 + 0.485380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16621 + 0.485380i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.278i)T \) |
| 43 | \( 1 + (-4.96 + 4.28i)T \) |
| good | 3 | \( 1 + (-2.24 - 0.882i)T + (2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (0.933 - 0.0699i)T + (4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (0.726 - 1.25i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.74 - 0.855i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 2.24i)T + (-4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.126 + 1.68i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-3.85 - 4.15i)T + (-1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.74 - 0.536i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.580 + 0.228i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-0.768 - 0.115i)T + (29.6 + 9.13i)T^{2} \) |
| 37 | \( 1 + (2.11 - 1.22i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.330 - 0.414i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (0.534 + 2.34i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-6.92 + 10.1i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (2.20 + 4.57i)T + (-36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-0.645 - 4.27i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (10.3 + 11.1i)T + (-5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (14.5 + 4.48i)T + (58.6 + 39.9i)T^{2} \) |
| 73 | \( 1 + (1.52 - 1.04i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-6.13 + 10.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.78 - 1.09i)T + (60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-0.921 + 2.34i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (2.12 - 9.31i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−11.76598868509698628315174773023, −10.34572441436624553820887265368, −9.449372553545702456564313147299, −9.052061656820672027883424841252, −8.144104313566862485716507271558, −7.27912288029862185594896496528, −6.07251309963696114199557338410, −4.01247535076932932859120097544, −3.24422358046811162643897745758, −1.85336016704403575818917898965,
1.19598977077189904737298363908, 2.77076403541441850296623124263, 3.85377640088255268584650625209, 5.94718790667007729288999344367, 7.08965292105115101789195855494, 7.71670448818250098291788651138, 8.623214210974945404651116582157, 9.217982010063307306255350344658, 10.22088161988141586974561933829, 11.29862227132955079253443160159
