| L(s) = 1 | + (−2.01 + 1.46i)2-s + (0.309 − 0.951i)3-s + (1.29 − 3.98i)4-s + (−1.44 + 1.70i)5-s + (0.768 + 2.36i)6-s − 7-s + (1.68 + 5.17i)8-s + (−0.809 − 0.587i)9-s + (0.422 − 5.54i)10-s + (−0.214 + 0.155i)11-s + (−3.38 − 2.46i)12-s + (1.25 + 0.914i)13-s + (2.01 − 1.46i)14-s + (1.17 + 1.90i)15-s + (−4.18 − 3.03i)16-s + (0.612 + 1.88i)17-s + ⋯ |
| L(s) = 1 | + (−1.42 + 1.03i)2-s + (0.178 − 0.549i)3-s + (0.647 − 1.99i)4-s + (−0.647 + 0.762i)5-s + (0.313 + 0.965i)6-s − 0.377·7-s + (0.594 + 1.83i)8-s + (−0.269 − 0.195i)9-s + (0.133 − 1.75i)10-s + (−0.0647 + 0.0470i)11-s + (−0.978 − 0.710i)12-s + (0.349 + 0.253i)13-s + (0.537 − 0.390i)14-s + (0.302 + 0.491i)15-s + (−1.04 − 0.759i)16-s + (0.148 + 0.456i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.220590 - 0.164735i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.220590 - 0.164735i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (1.44 - 1.70i)T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + (2.01 - 1.46i)T + (0.618 - 1.90i)T^{2} \) |
| 11 | \( 1 + (0.214 - 0.155i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 0.914i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.612 - 1.88i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.44 + 4.44i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.31 - 2.41i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.89 + 8.92i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.569 + 1.75i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (8.66 + 6.29i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (8.64 + 6.27i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 0.340T + 43T^{2} \) |
| 47 | \( 1 + (-0.0625 + 0.192i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.55 + 7.86i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.85 - 1.35i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.35 + 5.34i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.88 - 8.88i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.11 + 9.59i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.04 + 0.762i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.18 - 6.72i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.57 + 4.85i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.89 - 2.10i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.37 - 4.23i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.41333374254406230617932267006, −9.631516055630741816645699107945, −8.584372158599094643839534553466, −8.035759949381033307570404856984, −7.05966738438608329908088049558, −6.64333523859920131465325409606, −5.64545962452296862488832142177, −3.78982754694596639733502080474, −2.14873534536635771953439758402, −0.26257680604016125587411747956,
1.39981815650734797040103600553, 3.03075064964641000905748004614, 3.86946173341877953391778426156, 5.21736118340404029860476845697, 6.88333188778539109788281407668, 8.128112844371914322565103432782, 8.487362310089932875725245890518, 9.319836877693985146454983831889, 10.19916661422193497282023559075, 10.70367152121727443751091672747
