L(s) = 1 | + (2.28 + 0.262i)2-s + (1.65 + 1.20i)3-s + (3.22 + 0.750i)4-s + (−0.564 + 0.825i)5-s + (3.48 + 3.19i)6-s + (−1.38 − 1.42i)7-s + (2.84 + 1.01i)8-s + (0.373 + 1.14i)9-s + (−1.50 + 1.74i)10-s + (1.27 + 3.06i)11-s + (4.44 + 5.13i)12-s + (0.362 − 0.601i)13-s + (−2.79 − 3.62i)14-s + (−1.93 + 0.689i)15-s + (0.312 + 0.153i)16-s + (−3.78 + 3.09i)17-s + ⋯ |
L(s) = 1 | + (1.61 + 0.185i)2-s + (0.958 + 0.696i)3-s + (1.61 + 0.375i)4-s + (−0.252 + 0.369i)5-s + (1.42 + 1.30i)6-s + (−0.523 − 0.538i)7-s + (1.00 + 0.359i)8-s + (0.124 + 0.382i)9-s + (−0.477 + 0.550i)10-s + (0.385 + 0.922i)11-s + (1.28 + 1.48i)12-s + (0.100 − 0.166i)13-s + (−0.747 − 0.969i)14-s + (−0.498 + 0.177i)15-s + (0.0781 + 0.0384i)16-s + (−0.917 + 0.750i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.83894 + 1.90280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.83894 + 1.90280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (-1.27 - 3.06i)T \) |
good | 2 | \( 1 + (-2.28 - 0.262i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (-1.65 - 1.20i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (1.38 + 1.42i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (-0.362 + 0.601i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (3.78 - 3.09i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-6.59 - 0.377i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-1.25 + 8.70i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (-1.36 + 3.49i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (6.33 + 2.67i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (0.532 - 6.19i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (-4.81 + 2.71i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (8.40 - 2.46i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (0.497 - 2.45i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (7.36 - 3.61i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-5.17 - 2.92i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (9.89 - 1.13i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (-2.92 + 6.41i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.81 - 10.7i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (1.29 + 2.45i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (-0.184 - 6.44i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (-4.16 - 0.721i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (-12.0 - 7.76i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.78 - 9.91i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.86765413874944568791354457486, −9.951775365045132248287172514197, −9.149292970042953938016203750862, −7.949935835266791374072880597124, −6.89629198641682597243774996421, −6.26907262619288563732482401452, −4.82616239849800926523900607538, −4.08787103328883553183579170365, −3.41029162990582380429951176949, −2.46169285186405541009709533739,
1.75115195353113337680725966093, 3.10234855950294946858697304820, 3.46813739176609511046077855423, 4.98224025081167072192454040995, 5.72563974308409275135402415041, 6.85068483899752381600356600801, 7.60192442406510067592996704434, 8.893539154554120181344888389068, 9.310401444201869410044959885665, 11.12301133644216636363716600145