| L(s) = 1 | + (0.209 + 1.99i)2-s + (0.655 + 0.615i)3-s + (−1.96 + 0.417i)4-s + (2.90 + 1.44i)5-s + (−1.08 + 1.43i)6-s + (−1.33 − 2.55i)7-s + (−0.00478 − 0.0147i)8-s + (−0.137 − 2.18i)9-s + (−2.26 + 6.09i)10-s + (−0.564 − 2.40i)11-s + (−1.54 − 0.935i)12-s + (−4.62 + 0.388i)13-s + (4.81 − 3.19i)14-s + (1.01 + 2.73i)15-s + (−3.64 + 1.62i)16-s + (−0.461 + 0.0193i)17-s + ⋯ |
| L(s) = 1 | + (0.147 + 1.40i)2-s + (0.378 + 0.355i)3-s + (−0.981 + 0.208i)4-s + (1.30 + 0.645i)5-s + (−0.444 + 0.585i)6-s + (−0.505 − 0.966i)7-s + (−0.00169 − 0.00520i)8-s + (−0.0458 − 0.729i)9-s + (−0.716 + 1.92i)10-s + (−0.170 − 0.725i)11-s + (−0.445 − 0.269i)12-s + (−1.28 + 0.107i)13-s + (1.28 − 0.854i)14-s + (0.262 + 0.706i)15-s + (−0.910 + 0.405i)16-s + (−0.111 + 0.00468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.807009 + 1.20802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.807009 + 1.20802i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (9.57 + 7.69i)T \) |
| good | 2 | \( 1 + (-0.209 - 1.99i)T + (-1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (-0.655 - 0.615i)T + (0.188 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-2.90 - 1.44i)T + (3.02 + 3.98i)T^{2} \) |
| 7 | \( 1 + (1.33 + 2.55i)T + (-3.99 + 5.74i)T^{2} \) |
| 11 | \( 1 + (0.564 + 2.40i)T + (-9.85 + 4.89i)T^{2} \) |
| 13 | \( 1 + (4.62 - 0.388i)T + (12.8 - 2.16i)T^{2} \) |
| 17 | \( 1 + (0.461 - 0.0193i)T + (16.9 - 1.42i)T^{2} \) |
| 19 | \( 1 + (-0.488 + 1.50i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.41 + 1.56i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (1.43 - 7.52i)T + (-26.9 - 10.6i)T^{2} \) |
| 31 | \( 1 + (-3.48 + 2.76i)T + (7.07 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-5.11 - 7.35i)T + (-12.8 + 34.6i)T^{2} \) |
| 41 | \( 1 + (-9.60 - 1.21i)T + (39.7 + 10.1i)T^{2} \) |
| 43 | \( 1 + (3.42 - 6.55i)T + (-24.5 - 35.3i)T^{2} \) |
| 47 | \( 1 + (-2.81 - 6.70i)T + (-32.8 + 33.5i)T^{2} \) |
| 53 | \( 1 + (9.03 + 4.96i)T + (28.3 + 44.7i)T^{2} \) |
| 59 | \( 1 + (7.68 + 5.58i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (9.97 + 3.01i)T + (50.8 + 33.7i)T^{2} \) |
| 67 | \( 1 + (-0.175 + 2.79i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (-6.52 - 0.273i)T + (70.7 + 5.94i)T^{2} \) |
| 73 | \( 1 + (7.69 - 12.1i)T + (-31.0 - 66.0i)T^{2} \) |
| 79 | \( 1 + (-13.7 + 5.44i)T + (57.5 - 54.0i)T^{2} \) |
| 83 | \( 1 + (2.98 - 0.765i)T + (72.7 - 39.9i)T^{2} \) |
| 89 | \( 1 + (0.00333 + 0.00340i)T + (-1.86 + 88.9i)T^{2} \) |
| 97 | \( 1 + (-3.92 - 2.61i)T + (37.5 + 89.4i)T^{2} \) |
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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
−13.86852259270038672137838384549, −12.83582812839246607308016853768, −11.03793155172355679194223514212, −9.911753905963479312190936287362, −9.255783985933241011146895641645, −7.81109341190127243823254274071, −6.68141680732877680149557584114, −6.12800680276935238759155151852, −4.68713032914730600418716814223, −2.91719504272333759025888001663,
2.00311552171611740906040428307, 2.57615790914410202503143984950, 4.69181336354202367496984049391, 5.84167718953068578403900009700, 7.54951428722036660027723014789, 9.149855584451286945191830483872, 9.673426027788696111287249202796, 10.55053062231097822094916587077, 12.07640678450575047958149859889, 12.56178930872169778153515099326
