| L(s) = 1 | + (−0.647 + 1.27i)2-s + (0.0790 − 1.50i)3-s + (−0.0203 − 0.0280i)4-s + (0.543 + 2.16i)5-s + (1.86 + 1.07i)6-s + (2.39 − 2.96i)7-s + (−2.76 + 0.438i)8-s + (0.712 + 0.0748i)9-s + (−3.10 − 0.714i)10-s + (−1.28 + 2.88i)11-s + (−0.0439 + 0.0285i)12-s + (3.74 + 2.43i)13-s + (2.21 + 4.96i)14-s + (3.31 − 0.648i)15-s + (1.25 − 3.86i)16-s + (−0.617 + 1.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.457 + 0.898i)2-s + (0.0456 − 0.871i)3-s + (−0.0101 − 0.0140i)4-s + (0.242 + 0.970i)5-s + (0.762 + 0.440i)6-s + (0.907 − 1.12i)7-s + (−0.978 + 0.155i)8-s + (0.237 + 0.0249i)9-s + (−0.983 − 0.225i)10-s + (−0.387 + 0.870i)11-s + (−0.0126 + 0.00824i)12-s + (1.03 + 0.674i)13-s + (0.591 + 1.32i)14-s + (0.856 − 0.167i)15-s + (0.314 − 0.967i)16-s + (−0.149 + 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.926453 + 0.505512i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.926453 + 0.505512i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.543 - 2.16i)T \) |
| 31 | \( 1 + (5.11 + 2.21i)T \) |
| good | 2 | \( 1 + (0.647 - 1.27i)T + (-1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (-0.0790 + 1.50i)T + (-2.98 - 0.313i)T^{2} \) |
| 7 | \( 1 + (-2.39 + 2.96i)T + (-1.45 - 6.84i)T^{2} \) |
| 11 | \( 1 + (1.28 - 2.88i)T + (-7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-3.74 - 2.43i)T + (5.28 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.617 - 1.60i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.535 - 2.52i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (1.43 + 9.06i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.479 + 1.47i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.000891 + 0.00332i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.05 + 2.28i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (4.86 + 7.48i)T + (-17.4 + 39.2i)T^{2} \) |
| 47 | \( 1 + (-0.127 + 0.0650i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (5.35 - 4.33i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-1.76 + 1.58i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 1.40iT - 61T^{2} \) |
| 67 | \( 1 + (3.84 - 14.3i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.964 - 9.17i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (3.86 + 10.0i)T + (-54.2 + 48.8i)T^{2} \) |
| 79 | \( 1 + (-6.28 + 2.79i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.744 - 0.0390i)T + (82.5 - 8.67i)T^{2} \) |
| 89 | \( 1 + (-8.24 + 5.99i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (17.7 + 2.81i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.28816513006533497200097342225, −12.17788765363477917076736683336, −10.99115843340225243597195404967, −10.13533501909050354887804958619, −8.500038993357691964236514257567, −7.57829176079896542842800793260, −6.99854827295233944816378883294, −6.16310237248761912837964201346, −4.11742812136438537316907169305, −2.01152835082927512521959139936,
1.56220253839859539546635974181, 3.34070049134977330074879936340, 5.06737348727256817297143803271, 5.82876849603364224800979209732, 8.177027968318061238349773169648, 9.022863190997096068772027533952, 9.651597319161067737417698341720, 10.93565073823011576444688803348, 11.41691347071050385967649580105, 12.56269573118842145737523704018
