| L(s) = 1 | + (0.358 + 0.984i)2-s + (−0.523 − 0.0922i)3-s + (0.691 − 0.580i)4-s + (2.23 − 0.0926i)5-s + (−0.0966 − 0.548i)6-s + (−2.37 + 1.37i)7-s + (2.63 + 1.52i)8-s + (−2.55 − 0.929i)9-s + (0.891 + 2.16i)10-s + (−0.416 + 0.721i)11-s + (−0.415 + 0.239i)12-s + (0.601 − 0.106i)13-s + (−2.19 − 1.84i)14-s + (−1.17 − 0.157i)15-s + (−0.239 + 1.35i)16-s + (−1.65 − 4.54i)17-s + ⋯ |
| L(s) = 1 | + (0.253 + 0.695i)2-s + (−0.302 − 0.0532i)3-s + (0.345 − 0.290i)4-s + (0.999 − 0.0414i)5-s + (−0.0394 − 0.223i)6-s + (−0.896 + 0.517i)7-s + (0.930 + 0.537i)8-s + (−0.851 − 0.309i)9-s + (0.281 + 0.684i)10-s + (−0.125 + 0.217i)11-s + (−0.119 + 0.0692i)12-s + (0.166 − 0.0294i)13-s + (−0.587 − 0.493i)14-s + (−0.304 − 0.0407i)15-s + (−0.0598 + 0.339i)16-s + (−0.401 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11078 + 0.370714i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11078 + 0.370714i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.23 + 0.0926i)T \) |
| 19 | \( 1 + (4.35 - 0.175i)T \) |
| good | 2 | \( 1 + (-0.358 - 0.984i)T + (-1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (0.523 + 0.0922i)T + (2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (2.37 - 1.37i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.416 - 0.721i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.601 + 0.106i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.65 + 4.54i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.41 + 2.87i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (3.73 + 1.35i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.46 - 5.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.33iT - 37T^{2} \) |
| 41 | \( 1 + (-0.923 + 5.23i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.72 - 8.01i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.16 + 3.19i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-8.78 - 10.4i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-9.41 + 3.42i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.94 + 5.83i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.73 + 10.2i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.519 + 0.435i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (6.90 + 1.21i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.604 - 3.42i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.30 + 2.48i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.02 - 5.79i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.25 - 11.6i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−14.19007542566389157610018468116, −13.28115825300814417682977997992, −12.08808664417561710670984033256, −10.83915955573337868589985409674, −9.762145209819457298909434159449, −8.592001462815371084832356951861, −6.74944555780745769260838060983, −6.13827242286301481705831030173, −5.10891835690588552618061389957, −2.53120768778090713016460478818,
2.30160071333362868664815173823, 3.83829746287253239898675041780, 5.77937624631074079566457825309, 6.79625023744108769264490521577, 8.456751614302426042495454255788, 9.991982284195746435178616106519, 10.69311468000793510380211602178, 11.68515469735801165779234977588, 13.13363367604596883889822704696, 13.28828007219007026527794338031
