| L(s) = 1 | + (2.70 − 0.575i)2-s + (−1.37 + 2.17i)3-s + (5.18 − 2.30i)4-s + (−0.506 + 0.729i)5-s + (−2.48 + 6.67i)6-s + (−4.17 − 1.26i)7-s + (8.23 − 5.98i)8-s + (−1.54 − 3.27i)9-s + (−0.953 + 2.26i)10-s + (0.695 − 1.33i)11-s + (−2.13 + 14.4i)12-s + (0.750 + 0.372i)13-s + (−12.0 − 1.01i)14-s + (−0.885 − 2.10i)15-s + (11.2 − 12.5i)16-s + (−0.0210 + 0.0898i)17-s + ⋯ |
| L(s) = 1 | + (1.91 − 0.407i)2-s + (−0.795 + 1.25i)3-s + (2.59 − 1.15i)4-s + (−0.226 + 0.326i)5-s + (−1.01 + 2.72i)6-s + (−1.57 − 0.476i)7-s + (2.91 − 2.11i)8-s + (−0.513 − 1.09i)9-s + (−0.301 + 0.717i)10-s + (0.209 − 0.401i)11-s + (−0.615 + 4.16i)12-s + (0.208 + 0.103i)13-s + (−3.21 − 0.270i)14-s + (−0.228 − 0.543i)15-s + (2.81 − 3.12i)16-s + (−0.00511 + 0.0217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23000 + 0.245658i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23000 + 0.245658i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-8.29 - 9.06i)T \) |
| good | 2 | \( 1 + (-2.70 + 0.575i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (1.37 - 2.17i)T + (-1.27 - 2.71i)T^{2} \) |
| 5 | \( 1 + (0.506 - 0.729i)T + (-1.74 - 4.68i)T^{2} \) |
| 7 | \( 1 + (4.17 + 1.26i)T + (5.83 + 3.87i)T^{2} \) |
| 11 | \( 1 + (-0.695 + 1.33i)T + (-6.27 - 9.03i)T^{2} \) |
| 13 | \( 1 + (-0.750 - 0.372i)T + (7.85 + 10.3i)T^{2} \) |
| 17 | \( 1 + (0.0210 - 0.0898i)T + (-15.2 - 7.55i)T^{2} \) |
| 19 | \( 1 + (4.03 + 2.93i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.527 - 5.01i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (0.179 - 0.0459i)T + (25.4 - 13.9i)T^{2} \) |
| 31 | \( 1 + (-4.73 - 2.86i)T + (14.3 + 27.4i)T^{2} \) |
| 37 | \( 1 + (3.63 - 2.41i)T + (14.3 - 34.1i)T^{2} \) |
| 41 | \( 1 + (0.0372 + 0.0449i)T + (-7.68 + 40.2i)T^{2} \) |
| 43 | \( 1 + (6.78 - 2.04i)T + (35.8 - 23.7i)T^{2} \) |
| 47 | \( 1 + (-1.57 + 0.266i)T + (44.3 - 15.4i)T^{2} \) |
| 53 | \( 1 + (-6.27 - 2.48i)T + (38.6 + 36.2i)T^{2} \) |
| 59 | \( 1 + (2.37 + 7.29i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.98 + 0.250i)T + (60.7 - 5.10i)T^{2} \) |
| 67 | \( 1 + (0.175 - 0.372i)T + (-42.7 - 51.6i)T^{2} \) |
| 71 | \( 1 + (1.37 + 5.86i)T + (-63.5 + 31.5i)T^{2} \) |
| 73 | \( 1 + (-11.4 + 10.7i)T + (4.58 - 72.8i)T^{2} \) |
| 79 | \( 1 + (-1.91 - 1.05i)T + (42.3 + 66.7i)T^{2} \) |
| 83 | \( 1 + (-0.950 - 4.98i)T + (-77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (2.26 + 0.789i)T + (69.7 + 55.2i)T^{2} \) |
| 97 | \( 1 + (-0.765 + 0.0642i)T + (95.6 - 16.1i)T^{2} \) |
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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
−13.11852923122805915903763968217, −12.04251823109602334676593197063, −11.13599978856373195385438012818, −10.50293863776033191421723446814, −9.623929173620821760481023970851, −6.89156877141657021577032466987, −6.18374053513761673761839773468, −5.06323398878655815770375711437, −3.89461701533842053310871934658, −3.22091554864119734594014576627,
2.44120092718468827048743482346, 4.03220617322483686429541533175, 5.54313511799964282669459472566, 6.44975891518812787615238497818, 6.82371079502775658261479745044, 8.230546608163256725385084104546, 10.44347048825798420272958313438, 11.85342821173083232654000389919, 12.32721843951809210807880888820, 12.86605287461005551137317377682
