L(s) = 1 | + (1.92 + 1.73i)2-s + (0.994 − 0.104i)3-s + (0.490 + 4.67i)4-s + (0.450 − 2.19i)5-s + (2.09 + 1.52i)6-s + (−1.42 + 2.22i)7-s + (−4.10 + 5.64i)8-s + (0.978 − 0.207i)9-s + (4.65 − 3.43i)10-s + (2.32 + 0.494i)11-s + (0.976 + 4.59i)12-s + (1.90 + 0.618i)13-s + (−6.60 + 1.81i)14-s + (0.219 − 2.22i)15-s + (−8.47 + 1.80i)16-s + (−1.91 − 4.30i)17-s + ⋯ |
L(s) = 1 | + (1.35 + 1.22i)2-s + (0.574 − 0.0603i)3-s + (0.245 + 2.33i)4-s + (0.201 − 0.979i)5-s + (0.854 + 0.620i)6-s + (−0.538 + 0.842i)7-s + (−1.44 + 1.99i)8-s + (0.326 − 0.0693i)9-s + (1.47 − 1.08i)10-s + (0.701 + 0.149i)11-s + (0.281 + 1.32i)12-s + (0.528 + 0.171i)13-s + (−1.76 + 0.485i)14-s + (0.0566 − 0.574i)15-s + (−2.11 + 0.450i)16-s + (−0.464 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.19457 + 2.62009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19457 + 2.62009i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.994 + 0.104i)T \) |
| 5 | \( 1 + (-0.450 + 2.19i)T \) |
| 7 | \( 1 + (1.42 - 2.22i)T \) |
good | 2 | \( 1 + (-1.92 - 1.73i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (-2.32 - 0.494i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.90 - 0.618i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.91 + 4.30i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.305 - 2.90i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (6.04 + 5.43i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (5.97 - 4.33i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-8.80 + 3.91i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.434 - 2.04i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-0.548 + 1.68i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 5.86iT - 43T^{2} \) |
| 47 | \( 1 + (-4.58 + 10.2i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.14 + 0.435i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (1.40 + 1.56i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (8.73 - 9.70i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.30 - 7.42i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-0.106 + 0.0776i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.15 - 5.45i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (9.72 + 4.32i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-6.24 + 8.60i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (3.61 - 4.01i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (4.22 + 5.81i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.89560163281751965396480219072, −9.902426633931164180793306938823, −8.820202666073436890975726857700, −8.450050347139069849005943111203, −7.26170497629005247774916160955, −6.29979895581382348472531112434, −5.62644083191450190830021792783, −4.52038289612098949354207438951, −3.74129893750383825487675443607, −2.34967487732944305825469918123,
1.59873677413631449223604057537, 2.85772557944189283136525238302, 3.71236613360241549187653113459, 4.29631720974526690269746357636, 5.98675210649058270089095861839, 6.49825359772884706148551992259, 7.79267626159918497485074829064, 9.395080230812598834885220258037, 10.03040179548899149360459627593, 10.84841533267067990567327126453