| L(s) = 1 | + (1.92 − 0.168i)2-s + (0.456 + 1.67i)3-s + (1.68 − 0.297i)4-s + (−0.875 − 1.87i)5-s + (1.15 + 3.13i)6-s + (−2.22 − 0.810i)7-s + (−0.529 + 0.141i)8-s + (−2.58 + 1.52i)9-s + (−1.99 − 3.45i)10-s + (1.34 − 2.33i)11-s + (1.26 + 2.68i)12-s + (4.27 + 2.99i)13-s + (−4.40 − 1.18i)14-s + (2.73 − 2.31i)15-s + (−4.21 + 1.53i)16-s + (0.0267 − 0.0187i)17-s + ⋯ |
| L(s) = 1 | + (1.35 − 0.118i)2-s + (0.263 + 0.964i)3-s + (0.844 − 0.148i)4-s + (−0.391 − 0.839i)5-s + (0.472 + 1.27i)6-s + (−0.841 − 0.306i)7-s + (−0.187 + 0.0501i)8-s + (−0.861 + 0.508i)9-s + (−0.631 − 1.09i)10-s + (0.405 − 0.702i)11-s + (0.366 + 0.775i)12-s + (1.18 + 0.829i)13-s + (−1.17 − 0.315i)14-s + (0.706 − 0.598i)15-s + (−1.05 + 0.383i)16-s + (0.00648 − 0.00454i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77303 + 0.244911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77303 + 0.244911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.456 - 1.67i)T \) |
| 37 | \( 1 + (0.530 - 6.05i)T \) |
| good | 2 | \( 1 + (-1.92 + 0.168i)T + (1.96 - 0.347i)T^{2} \) |
| 5 | \( 1 + (0.875 + 1.87i)T + (-3.21 + 3.83i)T^{2} \) |
| 7 | \( 1 + (2.22 + 0.810i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.34 + 2.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.27 - 2.99i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.0267 + 0.0187i)T + (5.81 - 15.9i)T^{2} \) |
| 19 | \( 1 + (-0.263 + 3.01i)T + (-18.7 - 3.29i)T^{2} \) |
| 23 | \( 1 + (0.797 - 2.97i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.68 - 10.0i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (4.23 + 4.23i)T + 31iT^{2} \) |
| 41 | \( 1 + (-0.222 - 1.26i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.05 + 7.05i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.29 + 1.32i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.17 + 3.22i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (9.30 + 4.33i)T + (37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-3.96 + 5.66i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (0.834 - 2.29i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.47 - 10.1i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 0.950iT - 73T^{2} \) |
| 79 | \( 1 + (6.97 - 3.25i)T + (50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (-3.55 - 0.627i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.65 + 16.4i)T + (-57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 3.58i)T + (84.0 + 48.5i)T^{2} \) |
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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
−13.70324303264540624927727925117, −12.91587116019152703003033582182, −11.73812566949613728590838583589, −10.89294843149323270136751081351, −9.266781305291112098118130235254, −8.625628368961338254092862434594, −6.50914899529504140820251329650, −5.27130611762170251080976870088, −4.10567965302082345905137044946, −3.34224786641709923143799211211,
2.79500083516919956715460631185, 3.83613580422935742173360774575, 5.90227052216028559368000998758, 6.52751239205155387963565453638, 7.72434358695350962000725142796, 9.218550865489869071832361642695, 10.86152801342457739450958708213, 12.09595622657885349051778930810, 12.67611038582040004258757948858, 13.56651411118576554483117683138
