| L(s) = 1 | + (−0.822 + 1.42i)2-s + (0.695 + 0.505i)3-s + (−0.351 − 0.609i)4-s + (1.96 + 0.418i)5-s + (−1.29 + 0.575i)6-s + (2.79 + 0.595i)7-s − 2.13·8-s + (−0.698 − 2.14i)9-s + (−2.21 + 2.45i)10-s + (−0.195 − 1.85i)11-s + (0.0632 − 0.601i)12-s + (−0.659 + 6.27i)13-s + (−3.14 + 3.49i)14-s + (1.15 + 1.28i)15-s + (2.45 − 4.25i)16-s + (−3.01 − 3.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.581 + 1.00i)2-s + (0.401 + 0.291i)3-s + (−0.175 − 0.304i)4-s + (0.879 + 0.186i)5-s + (−0.527 + 0.234i)6-s + (1.05 + 0.224i)7-s − 0.753·8-s + (−0.232 − 0.716i)9-s + (−0.699 + 0.776i)10-s + (−0.0588 − 0.559i)11-s + (0.0182 − 0.173i)12-s + (−0.182 + 1.74i)13-s + (−0.841 + 0.934i)14-s + (0.298 + 0.331i)15-s + (0.614 − 1.06i)16-s + (−0.730 − 0.810i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.712607 + 0.840088i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.712607 + 0.840088i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (6.99 - 10.1i)T \) |
| good | 2 | \( 1 + (0.822 - 1.42i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.695 - 0.505i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-1.96 - 0.418i)T + (4.56 + 2.03i)T^{2} \) |
| 7 | \( 1 + (-2.79 - 0.595i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (0.195 + 1.85i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (0.659 - 6.27i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (3.01 + 3.34i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 + (-0.170 + 0.295i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.02 + 4.37i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.19 - 5.77i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-9.29 + 4.13i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (9.02 + 6.55i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.98 + 0.422i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (0.531 - 5.05i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (5.57 - 4.04i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 + (7.53 + 3.35i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.647 + 1.99i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (4.86 - 5.40i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (0.0731 + 0.225i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.325 - 1.00i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.70 - 11.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (2.60 - 0.553i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (2.53 + 2.81i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−13.89366088208130211749749279414, −12.12414552388998484358116653851, −11.27244820249108839698901310173, −9.761209505582311034543174712710, −8.917075134809891747995448244162, −8.321244965098276941831411148437, −6.79888106217592766178207124388, −6.10772583443050238925068583259, −4.50098305991459227905171673445, −2.44986214627739123681424001832,
1.64060645349952907976987739285, 2.65104011659766268358193676899, 4.83049050059434493333363971529, 6.15995492931772566138330641576, 7.990852019482218720499845385614, 8.563101490800975362604822820193, 10.01954205981824622711018273883, 10.52916190384363282127842543674, 11.47350292541448345968756906373, 12.82074577852691600978703643481
