| L(s) = 1 | + (−0.845 − 1.13i)2-s + (1.55 + 0.757i)3-s + (−0.568 + 1.91i)4-s + (−0.405 + 1.19i)5-s + (−0.459 − 2.40i)6-s + (0.538 − 0.702i)7-s + (2.65 − 0.977i)8-s + (1.85 + 2.35i)9-s + (1.69 − 0.551i)10-s + (1.72 + 0.113i)11-s + (−2.33 + 2.55i)12-s + (−0.136 + 0.155i)13-s + (−1.25 − 0.0166i)14-s + (−1.53 + 1.55i)15-s + (−3.35 − 2.18i)16-s + (−1.23 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.598 − 0.801i)2-s + (0.899 + 0.437i)3-s + (−0.284 + 0.958i)4-s + (−0.181 + 0.534i)5-s + (−0.187 − 0.982i)6-s + (0.203 − 0.265i)7-s + (0.938 − 0.345i)8-s + (0.617 + 0.786i)9-s + (0.537 − 0.174i)10-s + (0.520 + 0.0341i)11-s + (−0.674 + 0.737i)12-s + (−0.0378 + 0.0431i)13-s + (−0.334 − 0.00443i)14-s + (−0.397 + 0.401i)15-s + (−0.838 − 0.545i)16-s + (−0.300 − 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40342 + 0.284873i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40342 + 0.284873i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.845 + 1.13i)T \) |
| 3 | \( 1 + (-1.55 - 0.757i)T \) |
| good | 5 | \( 1 + (0.405 - 1.19i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (-0.538 + 0.702i)T + (-1.81 - 6.76i)T^{2} \) |
| 11 | \( 1 + (-1.72 - 0.113i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (0.136 - 0.155i)T + (-1.69 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.23 + 1.23i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.64 - 8.27i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.848 + 0.650i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-3.48 + 7.06i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-1.77 - 1.02i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.88 - 9.47i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (5.14 - 3.94i)T + (10.6 - 39.6i)T^{2} \) |
| 43 | \( 1 + (-0.193 + 2.95i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (-2.26 + 8.46i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.236 + 0.354i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (0.603 - 1.77i)T + (-46.8 - 35.9i)T^{2} \) |
| 61 | \( 1 + (-1.29 - 0.636i)T + (37.1 + 48.3i)T^{2} \) |
| 67 | \( 1 + (0.128 + 1.95i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (3.07 + 7.42i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-6.36 + 15.3i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (10.1 + 2.71i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-2.23 + 0.758i)T + (65.8 - 50.5i)T^{2} \) |
| 89 | \( 1 + (-8.28 + 3.42i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-3.16 + 1.82i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.42786354093341878437512218372, −10.10372227287310600268393627034, −9.102704083680267246060908904602, −8.268735595497823637095967024406, −7.62690836213024755820608222286, −6.54703891133829676337874928078, −4.67708924403381084437553611197, −3.80200131682920065735657602754, −2.90645151494299361826135857901, −1.64748295850038932511980984019,
1.00539999034171092251034345024, 2.46493183256066971086162926822, 4.17655296872074926031652868882, 5.15620037766837337688347342515, 6.53808454250865900895033829740, 7.11377725294425060503037116533, 8.175853214475412044377118669775, 8.901690940947789411817252605041, 9.200447358526014988131083406766, 10.44391100057400895806201447394
