| L(s) = 1 | + (−0.0523 − 0.998i)2-s + (2.57 + 2.08i)3-s + (−0.994 + 0.104i)4-s + (0.713 + 2.11i)5-s + (1.95 − 2.68i)6-s + (−2.57 − 0.621i)7-s + (0.156 + 0.987i)8-s + (1.66 + 7.83i)9-s + (2.07 − 0.823i)10-s + (1.27 + 0.269i)11-s + (−2.78 − 1.80i)12-s + (−2.17 − 4.27i)13-s + (−0.486 + 2.60i)14-s + (−2.58 + 6.95i)15-s + (0.978 − 0.207i)16-s + (1.76 + 0.678i)17-s + ⋯ |
| L(s) = 1 | + (−0.0370 − 0.706i)2-s + (1.48 + 1.20i)3-s + (−0.497 + 0.0522i)4-s + (0.319 + 0.947i)5-s + (0.796 − 1.09i)6-s + (−0.971 − 0.235i)7-s + (0.0553 + 0.349i)8-s + (0.555 + 2.61i)9-s + (0.657 − 0.260i)10-s + (0.382 + 0.0813i)11-s + (−0.803 − 0.521i)12-s + (−0.603 − 1.18i)13-s + (−0.129 + 0.695i)14-s + (−0.667 + 1.79i)15-s + (0.244 − 0.0519i)16-s + (0.428 + 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76774 + 0.704964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76774 + 0.704964i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0523 + 0.998i)T \) |
| 5 | \( 1 + (-0.713 - 2.11i)T \) |
| 7 | \( 1 + (2.57 + 0.621i)T \) |
| good | 3 | \( 1 + (-2.57 - 2.08i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.27 - 0.269i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (2.17 + 4.27i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 0.678i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.222 + 2.12i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-6.65 + 0.348i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (3.55 + 4.89i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.04 - 2.33i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-3.68 + 5.66i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-4.11 - 1.33i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.60 - 1.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.71 + 4.46i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-6.17 + 7.62i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (8.60 + 9.55i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-3.08 - 2.77i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.24 + 3.24i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (6.06 - 4.40i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.84 + 3.79i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (2.35 - 5.27i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (13.0 - 2.06i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (9.61 - 10.6i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (6.59 + 1.04i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.11840287089388774314173524346, −10.44705913777320133711555777441, −9.705705882851590122529551440404, −9.324834111831860461647685308752, −8.088514119477098938426889178314, −7.10619775990749856001173607419, −5.40513665918379540679872643220, −4.01948820204222645378293619318, −3.14790009291321169065853183476, −2.53994535875052177370022880701,
1.34649162358219603880464549690, 2.86156667534093542470190051981, 4.19977704147311910766119723121, 5.86168238610071134439902161705, 6.84298687319597798394969356356, 7.52397634900131120827742820080, 8.662255601260653597800646313840, 9.194702944459613539376497541125, 9.705410669604711767353959526029, 11.93797060446549589390414672937
